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ELEMENTS 



OF 



TRIGONOMETRY, 



PLANE AND SPHERICAL. 



iDAPTED TO THE hlESENT STAT£ OF At^ALTSfS. 



Td WHICB 18 ADJ)JSD, THETR APi^LICATlON TO TUE PRINCIPLF^ Of 

NAVIGATION AND NAUTICAL ASTRONOMY. 



WITH 



LOGAfelTriMiC, TRIGtO^OMETRICAL, AND NAUTICAL 

TABLES. 




• ■ • • • • 






liy^-§:^l/>\ USE or tOLLElaEti ^T^/^^AD&^tB, 



.VI.; xv: v^>^^^ 



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• •* *•> •»• 



• * 






6t TriE Rev. C. W. HACKLEY, 

ProfcnBr of MtUhimatiet in the Univernty of the City of NeteTork. 



• - - -—' *t v^ >/ 



PUBLISHED BY 



WILEY & PUTIfAii; COLLINS, KEEISE & Co., New-York— THOMAS, GOV 
THWAITE A Co., Philaoilphia— PERKINS & MARVIN, Boston— 
CUSHING Sc SONS, Baltimors— S. BABCOC& Sc Co., 
CHARLUTOif— SMITH Sc PALMER, Richmo. — 

1838. 



EnteNd according to Act of Congress, A. D. 1838, by Rev. C. W. Hacklky, 
in the Clerk's office of the District Courl of the United States for the 
Scat hem District of New-York. 



..-• .*VW ^ v-V 



• • • - 



• -^» 






NE"W-Y0RK : 

PRINTED BY WILLIAM OSBORN, 

88 WILLIAM-STREET. 



PREFACE. 



Analytical Trigonometry has always been to the 
majority of college students a dry and difficult study. 
A conviction that it might be rendered easy and in- 
teresting to all who have a tolerable acquaintance 
with Algebra and Geometry, has led to the produc- 
tion of the present work. The faults of former 
treatises on this subject, which have detracted from 
their usefulness as booka of instruction, appear to be 
these : .......... 

1. A too sudden transition ft^ (jredmetry to Tri- 
gonometry, in consequence of whir^b, the first efforts 
of the learner are in th^'rfaffk'(i^ to the object of his 
pursuit. 

2. A tedious succession of general formulae at the 
commencement, the use and application of which is 
so long delayed as to produce weariness and dis- 
couragement before there is any apparent fruit tr 
ward labor. 

3. Too much abridgment in the d^-^onstn 
and particularly in the derivation ff 
results. 



ly PREFACE, 

The author is awarie of the importance attached 
to the exercise of intellect required to discover the 
connexion between propositions whose mutual de- 
pendence is shown by intermediate links which th^ 
mind must supply unoided^but it will be admitted on 
th@ other hand, that the ordinary term of study is 
too limited, and the field of knowledge in this de- 
partment too extensive, to afford the loss pf tme 
which such a mode occasions. Besides, there wil] 
be abundant scope for this kind of exercise, in 4 
more mc^tured familiarity with mathematical reason- 
iilg, for which the shortening qf labor here, wil) 
Ipave additional rooip. 

The following work begins with some construcr 
tions of triangles ^ccordin^ to the rules given in ge- 
ometry, fdlld^ft.d' by oliCTSuir which scales of equal 
parts and protrsictors/are^ployedy showing at once 
and distinctlyjl'^ilt ig'tft jbe. understood by the solur 
^on of a triab'^e/ftwl: tfiqe .valije of trigonometry in 
the measurement of inaccessible heights and dip- 
(ances. 

The evident inaccuracy \n the use of instruments, 
leads the learner to perceive the necessity of a more 
exact and certain method, and prepares him to enter 
with satisfaction upon the study of Analytical Trigo- 
iiometry, 

explanation of the Trigonometrical Lipes has 
\ labored with great care, aqd it is believed, 



PREFACB. - V 

tbat considerable improTement in the method of ex^ 
hibiting their changes will be obsenred. Their a(H 
plication to the solution of triangles, is immediately 
shown in a few cases, with the help of a table of na-^ 
tnral sines and cosines at the end. 

Then follows a fall exposition of the theory and 
use of logarithms, with every variety of example. 

Part I. concludes with the application of logarithms 
and logarithmic sines, tangents, &c., to a number of 
practical examples involving every case in the solu-^ 
tion of plane triangles. 

Fart lit contains Spherical Trigonometry. Parti^r 
onlar care has been tf^cen to render the demonstra*^ 
tions here, plain and easy, and to avoid all unneces^ 
sary repetition and complication. 

It was found that the introduction of a few celes- 
tial circles, such for the most part as the study of 
geography may be supposed to have already rendered 
familiar, would afford ^^ opportunity for making all 
the examples of Spherical Trigonometry, Astrpnq? 

micc^lr 

This, together with the practical character of those 
in plane trigonometry, most of which are problems 
in the measurement of heights and distances, ' 
peculiar feature in the plan of the present m 
The copsideration which led to it was. that i 
trigonometry had grown out of the vai 

men in these very particulars, if th< pi 



W PREFACE. 

interesting to stimulate discovery, they would also 
the study of what is already known. The analytic 
method, though not always practicable before the 
mind is somewhat furnished, is doubtless, by far the 
best method of training. Besides this general reason 
for introducing astronomical problems here, it was 
deemed useful thus to prepare the way for the study 
of astronomy, whilst the formulse and rules of trigo- 
nometry were fresh in the memory, and to prevent 
that neglect of the trigonometrical solutions of as- 
tronomy, which is apt to result from the trouble of 
recalling what has been long laid aside. It was 
thought, too, that this foretaste of astronomy might 
excite a relish for that study. 

Part III. is a recapitulation of the formulae and rules 
demonstrated in the previous parts, collected for con- 
venience of reference or committing to memory. 

Part IV. exhibits the application of trigonometry to 
the principles of navigation and nautical astrono- 
my, and is chiefly from the admirable treatise of Mr. 
Young. This affords a pleasing and useful exercise 
in the formulae and rules of the preceding parts. . It 
will occupy the student but a few days, and has al- 
ways been considered an essential part of a po- 
lite education. Should it, however, be omitted, it 
will be necessary, previously to the study of Analytical 
imetry, to go over the Addenda contained in 
V. 



PREFACE < til 

Part VL is a sort of supplement which the general 
student may well omit, but which will be found to 
contain matter useful to practical men, and interest- 
ing to those more exclusively devoted to the math* 
ematics. 

It was the author's intention to have added some 
formulae applicable to geodesy and other branches 
of practical science, but considering that these are 
always demonstrated in the treatises upon the sub- 
jects to which they refer, it was thought advisable 
not to increase the size of this work, already suffi- 
ciently large fpr its main object, viz., the use of col- 
leges and the purposes of a general education. A 
valuable collection of logarithmic, nautical and as- 
tronomical tables, will be found at the end. 

The minute exhibition of the processes in the nu- 
merical examples has been regarded by the author 
as essential to the imparting of an available know- 
ledge of trigonometry. These examples are few in 
the first and second Parts, but the deficiency is abun- 
dantly supplied in Part IV., where, since they could 
be made most useful and interesting, it was thought 
that they might best be multiplied. 

In the preparation of the present work nothing has 
been sacrificed to the desire of originality. Free use 
has been made of the comprehensive treatise of Mr 
Young. Recourse has been had for some minor po 
tions, to the recent work of M. Francoeur upon Gc 



irtti pRErACE. 

de8y« The idea of the simple instrument for taking 
angles, described at Art^ 10, was derivcid ftoni a small 
work on geometry for beginners, by D r. Ritchie, late 
of the London University, The greater part of Ai*- 
ticle 130 is from the Edinburgh Encyclopedia* 

Ntw- York, May, 1838. 



CONTENTS. 



PART I. 



PLANE TRIGONOMETRY. 

Article p^gi 

1. Definition of Trigonometry, ----- i 

2. Data by which triangles are determined, - - - 1 

3. Geometrical construction of triangles, - - - - 2 

4. Scales of equal parts, ------ 3 

MEASUREMENT OF AN0L£8. 

5. By the sexagesimal division of the circumference, •» * - 3 

6. By the centesimal division, - - . «. ^ 6 

7. Engineers' method, -----*-7 

8. Protractors, ---.-*.- 7 

9. Use of the protractor, - . - . - - g 

SOLUTIONS BY CONSTRUCTION. 

10. Finding the distance of inaccessible objects, - . - £1 

11. Measurement of heights, - - • - • -11 

12. General view of Analytical Trigonometry, - - - 13 

TRIOONOMETRICAL LINES. 

13. Their use, --..--,- 13 

* THE SINE. 

14. Its definition, -------14 

15. Sine of the supplement, - - - • - - 14 

16. Variations in the value of the sine, - - - - 15 

17. Sines of other quadrants derived from the first, - - - 16 

18. Algebraic sign of the sine, - , - - - - 17 

THE TANGENT. 

19. Its definition and changes of value, - - - - - 18 

20. Recapitulation of the values of the tangent, - - * 20 

THE SECANT. 

21. Definition and variations of value of the secant, -> -^ •» 20 

22. Algebraic sign of the secant, ----- 31 

23. Definition of the complement of an arc or angle, - - - 21 



THE COSINE. 



24. Its definition, 

25. Its Algebraic sign, 
36. Its vanations of value, 
87. Negative arcs. 



X ' CONTENTS. 

THB COTANOEHT AND COSECANT. 

Article Pnf« 

28. Values of the cotangent, . . - > * - 24 

39. Values of the cosecant, - - - - - - 24 

30. Their Algebraic signs, ------ 24 

31. Algebraic notation of the trigonometrical lines, - - - 25 

32. Expression for the tangent in terms of the sine and cosine, - 25 

33. Expression for the secant, ------ 26 

34. " for the cotangent, ----- 27 

35. " for the cosecant, ------ 27 

36. Corollaries from these, - - . - - - 27 

37. Relation of tangent and cotangent, - - - - - 28 

FORMULA FOR THE SOLUTION OF RIGHT-ANGLED TRIANGLES, 

38. Derivation of formulae, ----- • 28 

39. 40, and 41. Applications of these formulae, - - - - 31 

42. Advantage of logarithms, - - - , - 35 

THEORY OF LOGARITHMS. 

43. Definition of logarithms, - - - - - - 35 

44. Logarithms of uie base and unity, .... 36 

45. DeSnition of a table of logarithms, - - - - - 36 

46. Method of calculating tables of logarithms, ... 37 

47. Theory of the characteristic, - - - - - 39 

Explanation of the Tables. 

48. Rule to find the logarithm of any number between 1 and 10,000, 40 

49. Multiplication and division by logarithms, - - - - 41 

50. Logarithms of decimal numbers, ----- 43 

51. Rule to find the logarithms of numbers greater than 10,000, - 45 

52. Rule to find the number corresponding to any given logarithm, 47 

53. Examples in multiplication and division by logarithms, - - 49 

54. Formation of powers by logarithms, - ... 51 

55. Extraction of roots by logarithms, - - - - - 52 

56. Table of logarithmic sines, tangents, &c., - - - 53 

57. Rule to find from the table the logarithmic sine, tangent, &c., of 

any given number of deuces, minutes, and seconds, - - 53 

58. 59. To find the degrees, minutes, and seconds corresponding to any 

given logarithmic sine, tangent, &c., - - - - 56 

60. Rules to find the logarithmic secant and cosecant of any given arc}- 57 

Solution of Hght-angled triangles with the aid of logarithms. 

61. Examples, --------59 

62. Use of the arithmetical complement, - ... 60 

63. Elxample in the measurement of distances, - - - - 61 

Solution of oblique-angled, triangles by logarithms. 

64. A side and the opposite angle being two of the given parts, - 62 

65. Two angles and the interjacent side being given, - - - 64 

66. Example in the measurement of heights, the bases of which are in- 

accessible, -------65 

67. Two sides and the angle opposite one of them being given ; an am- 

biguous case, -------66 

68. Derivation of a formula for the cosine of an angle in terms of the 

three sides of a triangle, - - • - - - 67 
I 70. Derivation of formulae for the sine and cosine of the sum 

id diflference of two arcs. - - - - - ^^ 69 
ivation of formula for the sine and cosine of an arc in terms of 
Alf the arc, - - -- - - -72 



CONTENTS. XI 

Article Pafe 

73. Derivation of a formula for the sine of half an arc in terms of the 

cosine of the arc, - - - - - - -73 

73. Derivation of a formula for the ane of half an angle in terms of the 

three sides of a spherical triangle, and example of its application, 73 

74. Derivation of formulae for the sum and difference of the sines of two 

arcs and the ratio of these, - - - - - 76 

75. Two sides and the included angle being given, the method of solu- 

tion, with examples, - - - - - - 78 



PART 11. 

SPHERICAL TRIGONOMETRY. 

77. Definitions and general principles, - - - - 81 

78. The hypothenuse of a right-angled triangle, being either given or 

required, - - - - - - - -82 

79. Definitions of celestial circles, ... * . 83 

80. Astronomical example under the last case of solution, - f- 85 

OBLIQUE ANQLED TRIANGLES. 

81. Two of the three given parts being a side and its opposite angle, 86 

82. Formula for the cosine of an angle in terms of the three sides, - 88 

83. Formula for the difference of the cosines of two arcs, - - 91 

84. Three sides of a triangle being given to find the angles, with an 

astronomical example, - - - - - -91 

85. Three angles being given to find the sides, »• - - 94 

86. Formulae for the sum of the cosines of two arcs, their ratio to the sura 

of the sines, to the difference of the sines, that of the sine of the sum 
to the sum of the sines and that of the sine of an arc to radius 
minus the cosine, - - - - - - -96 

87. Napier's analogies with astronomical examples, - - 98 

aiGHT ANGLED TRIANGLES. 

88. Napier's rules for the circular parts, .... 105 

89. 90 and 91. Astronomical examples, - - - - 107 

92. Example, given the sun's declination to find the time of his rising 

or setting, with explanation of the difference between mean and 
apparent time, - - - - - - -110 

93. Astronomical example, - - - - - 112 

94. Last case in the solution of oblique angled triangles, - - 113 



PART III. 

Recapitulation, ..-.-.. 

PART IV. 

APPLICATION OF PLANE AND SPHERICAL TPTnoNOM! 
TO THE PRINCIPLES OF NAVIQ/i > 

NAUTICAL ASTRONOW 

95. Introduction, - - . - 



XII CONTENTS. 

Chap. 1. Principles of Navigation. 

Article Paga 

96. Definitions, --*..__* l25 

97. Plane sailing, --*..-- 129 

98. Traverse sauing, ------ 133 

99. Parallel sailing, 138 

100. Middle latitude sailing, 141 

101. Mercator's sailing, --*---- 145 

102. Example of the last, - - . * - - 147 

Ciup. 2. Nautical Astronomy. 

103. Introductory, ------- 149 

104. Definitions, -------150 

105. Corrections to he applied to th£ observed altitudes of celestial objects^ 152 

106. Dip or depression of the horizon, ----- 153 

107. Semidiameter, ..*.--- 153 

108. Refraction, - - - - - - -155 

109. Parallax, 157 

110. Examples of corrections, -..---- 168 

111. Method of determining the latitude at sea by the meridian altitude, 161 

112. By two altitudes, - - - -- - -166 

113. On finding the longitude by the lunar observations, - ^ 171 

114. Variation of the compass, ------ 177 



PART V. 

ADDENDA. 

115. Formulae for the tangent of the sum and diffefcnice of two ares, 180 

116. Value of the sine and cosine of 45^, - - - - 181 

117. Sin i an arc = i chord of the arc, ... - 181 

118. Sine of 30«» and secant of 60°, ----- 182 

119. Tangent of 45«> ± an arc, - - ^ ^ - * 182 

120. Generallization of formulae for the sine of the sum df two arcs, 182 

121. Construction of table of sines, &c., - - - . 183 



PART VI. 

MISCELLANEOUS TRIGONOMETRICAL INaUIRIES. 

122. Introductory, ------- 185 

123. Solution of certain cases of plane triangles and the trigonometrical 

lines of small arcs, ------ 185 

124. Cluadrantal triangles, ------ 195 

125. Formulae to be employed instead of Napier's rules in certain cases 

where great accuracy is required, - . - - 199 

15%. Two sides and the included angle of a spherical triangle being given 

to find the third side directly, - - - - - 201 

127. Two angles and the interjacent side being given to find the third 

angle, - - - - - - - - 202 

128. Rules relative to ambiguous cases, - - - . 203 
Additional formulae wnere three sides or three angles of a spherical 

triangle are given, ------ 204 

determination of tho effect of minute errors in data, - - 205 



PART I, 



PLANE TRIGONOMETRY. 



1. The term trioonombtbt is compounded of two Greek 
words Tij^Y^yog a triangle, and inejQOP measure, signifying lite- 
ratly the measurement of triangles. It has for its object to 
determine the unknown parts of a triangle when a sufficient 
number of parts are known. 

By parts of a triangle are understood commonly the sides 
and angles, though trigonometry properly includes the mea- 
surement of the surface also. 

There will accordingly be six parts of every triangle, name- 
ly the three sides and the three angles. 

2. It has been proved, (Geom. *B. 1, Props. 6, 6, and 10,) that 
when two triangles have three parts, one of which is a side, 
in the one equal respectively to the corresponding parts in the 
other, the triangles are equal. 

One part must be a side, because if the three angles only 
were equal respectively in the two triangles they would be 
but similar, (Geom. B. 4, Prop. 18,) that is alike in shape bnl 
not necessarily in size. 

Since all triangles which have three parts equal, u 
consequence equal, it is said that three given parts detei 
a triangle, that is with these three given parts but one tH 
can be formed. 

If any number of attempts be made w ti 



* The Geometry to which we refer here Mid ell i u 

Legendre. 



2 



PLANE TRIGONOMETRY. 



with the same three given parts, the result will be always a 
repetition of the same triangle. The corresponding sides of 
the successive ones will not differ in length, and the angles 
will not differ in magnitude. 

There is one exception to this principle, pointed out in B. 3, 
Prob. 11, Geom., where two sides and the angle opposite 
one of them are given, in which case two triangles can be 
constructed with the given parts. 

3. Three parts of a plane triangle being given then, (ex- 
cept they be the three angles,) it ought to be possible to find 
the other three, since these are fixed by their dependance 
upon the three given. 

This may be accomplished with sufficient accuracy for 
many purposes, by means of constructions, such as are exhi- 
bited at problems 8, 9 and 10 of the 3d book of Geometry. 

We shall repeat one of these constructions, enunciating the 
problem somewhat differently. 

The two sides and included angle of a triangle being given, 
let it be required to find the remaining side and the other two 
angles. 

A B 

Let A and b be the two 
given sides, and c the gi- 
ven included angle. Draw 
two lines dh and g of 
indefinite length, making 
with each other an angle 
equal to the given angle 
c. Lay off on the first of 

these the given line a ^ e 

from D to E, and on the second the given line b from d to f. 
Join £F. The onlji; possible triangle def will thus be formed 
with the three given parts, in which ef will be the required 
aide, and e and f the required angles. 

The finding the unknown parts of a triangle by means of 
those which are given, is called its solution. 

4." The method of solution just exhibited is rendered more 
ctically useful by the employment of scales of equal parts 
1 protractors. 




THEORY OF SCALES. 3 

The most simple form of a scale of equal parts, is shown 
in the annexed figure. 

i \ \ \ \ \ \ \ \ In.rtmtJ 

It is a straight rule divided into any number of equal parts: 
in this example ten and one of these again into ten, so that 
the smallest division isone hundredth of the whole length of 
the rule. 

The following is the manner of usingit. 

Suppose that it is required to draw upon paper a line 
equal in length to 56. 

Place one foot of a pair of 
dividers at the line of division 
marked 6, and extend them till 
the other foot reaches exactly to 
the sixth smaller division mark 
on the right of ; the feet of ^ 
the dividers will then be at a distance of 56 apart. To draw 
now the required line upon paper, let a be the point from 
which it is to be drawn. Placing one foot of the dividers at 
A, extended the distance 56 obtained from the scale, describe 
with the other an arc of a circle on the side towards which 
the line is to be drawn ; then from a draw the line in the 
proper direction, terminating it at the arc before described, 
and it will be the line required. 

Another line of 42 being measured from the scale and laid 
down upon the paper, the two lines will be in the ratio of 56* 
to 42. If they are lines upon a map, and the first corresponds 
to a line of 56 feet upon the ground, the second will corres- 
pond to a line of 42 feet. 

If the first represent 56 yards, or chains, or miles, the se- 
cond will represent 42 yards, or chains, or miles. And in 
general lines upon the same drawing which are measured in 
parts of the same scale must be understood to be expressed * 
units of the same kind. 

5. Before describing the protractor which is an instrun 
for laying off angles, it will be necessary *' 'ain 
method of estimating the magnitude of angli 



4 PLANE TRIGONOMETRY. 

In Geometry (B. 3, Prop, 17. Schol. 1,) it is shown that 
angles are proportional to the arcs included between their 
sides, the arcs being described with equal radii, and it is also 
there stated that hence such arcs are properly the measures 
of angles. 

So that if an arc included between two sides of oue angle 
be double, or triple, or sextuple, an arc described with the 
same radius included between the sides of another angle, the 
first angle is double, triple, or sextuple the second. 

The relative magnitude of angles may therefore be cor- 
rectly expressed by means of the relative magnitudes of the 
arcs which measure them. 

The relative magnitudes of quantities are commonly given 
by referring the quantities to be compared to some known 
standard of measure which must be always of the same kind 
with the quantities themselves. 

This standard is called a unit. Thus a foot, a yard, &,c. 
are units of length, and the idea of the relative lengths of two 
lines is obtained by its being said that one is seven feet or 
yards, and the other nine. Or the just conception of the length 
of a single line is had by being told how many feet, yards or 
miles it contains. The mind compares it with one of these 
well known units, which in imagination it repeats along its 
length. 

Now the unit of measure, which is employed in a similar 
manner for giving the conception of the magnitude of an arc, 
is called a degree. A degree is the jU part of the circumfer- 
*ence of a circle. The relation which any given arc bears to 
the whole circumference may be conveniently expressed by 
stating the number of degrees which the arc contains. Thus 
an arc of 90 degrees will be one fourth the whole circumfer- 
ence. An arc of 45 degrees will be one eighth. An arc of 30 
degrees will be somewhatJess. And it is plain that the length 
of the arc, as compared with the whole circumference, may be 
readily conceived, as soon as the number of degrees which 
it contains is mentioned. 



MEASUREMENT OF ANOLBS. 




So also the magni- 
tude of the angles sub- 
tended by these arcs 
will, after a little fami- 
liarity, be rendered 
easily sensible to the 
mind. To speak of 
an angle of 10 degrees 
for instance, (A C B 
in the annexed dia- 
gram,) will suggest the 
image of a very acute 
angle, one of 60 degrees (A C D) a much larger acute angle, 
one of 140 degrees (A C E) an obtuse angl«. 

A degree being always the ^ij part of a circumferencci a 
single degree will be larger in alarger circle than in a smalleii 
and this, so far from being inconvenientf is particularly ad- 
vantageous in the measurement of angles ; for since arcs 
described about the vertex of an angle as a centre with different 
radii, and included between the sides of the angle, bear tbo 
same relation to each other as the radii, ^Geom. 6. 6, Prop, 11, 
Cor>) and since the entire circumferences are also proportional 
to their radii, it follows that two concentric* arcs included be- 
tween the sides of the same angle, and having the vertex of 
that angle for a centre, are the same aliquot parts of their 
respective circumferences. Consequently, two such arcs 
will contain the same number of degrees. Hence, to find the 
number of degrees contained in a given angle, the arc 
described for the purpose about the vertex, and extending 
from side to side of the angle, may be with any radius at 
pleasure. 



* Having the mme centre. 




6 PLANE TRIGONOMETRY. 

This may be distinctly seen in the following diagram. 

ACS is the angle '; the 
larger arc ab included 
between its sides contains 
50 degrees of the whole 
circumference ; the are 
ab with the lesser radius 
also contains 50 degrees, 
and so would an arc in- 
cluded between the sides 
of the given angle de- 
fK^ribed with any other 
radius whatever. 

Where the size of an angle is such that it does not embrace 
an exact even number of degrees of the circumference, smal- 
ler divisions called minutes, 60 of which make a degree, are 
employed. The angle is then said to contain as many de- 
grees and nrinutes as there are degrees and parts of a degree 
each, gJj, over, between its sides. If the second side of the an- 
gle does not pass exactly through one of these smaller divi- 
sions, a still smaller kind termed seconds, 60 of which form 
a minute, or 360 a degree, must be introduced. More minute 
divisions than these last are seldom used. When it becomes 
necessary to regard such, the same system is continued. The 
next denomination is thirds, 60 of which make a second, the 
next fourths, and so on. 

The notation for these denominations is as follows. De- 
grees are written thus^ ; minutes thus ' ; seconds thus " ; thirds 
thus '", &c. ; 30^ 20' 10" is read thirty degrees, twenty minutes 
and ten seconds. 

6. It is evident that the numbers used in the system of di- 
vision for the circumference of the circle, are entirely arbi- 
trary. Others might be employed with equal propriety, pro- 
vided the same principle were observed. In fact the attempt 
has been made, and probably will be successful in France, to 
subvert the old system of division, and to adopt a decimal 
system in this as well as in every other sort of measurement 
Thus a right angle, which is the unit of angles, is made to 



MEASUREMENT OF ANQLBS. 7 

contain 100^ instead of 90 ; and the circumference will then 
contain mP instead of 360. 100' instead of 60 ^l^ 100" -1'. 
The convenience of a decimal division we have experienced 
in this country in our system of Federal money. The French 
are likely, despite the despotism of custom, to enjoy the same 
advantage in all denominate numbers. 

7. Another method of expressing the magnitude of angles w 
as follows* 

A distance at pleasure is laid off from the vertex of the 
angle upon one of the sides, and a perpendicular there drawn 
to this side till it meets the other side of the angle. The 
ratio of this perpendicular to the distance from its foot to the 
vertex, serves to indicate the size of the angle. 

For example, if the line 
BCDE be perpendicular to 
the line ab, and bc be one 
fourth AB, the angle bag is 
said to be an angle of ^. If 
bd be one half ab, the angle 
bad is said to be an angle 
of ^. If be be equal to ab, 
BAE is said to be an angle of 
1; and so on for other mag- 
nitudes. An angle of 1 is 
plainly half a right angle, or 45^. 

This kind of measurement is much used by engineers, to 
express the degree of slope in excavations and embankments. 

8. The protractor which we are now prepared to describe, 
is an instrument for drawing upon paper an angle of any 
given n umber of degrees. 

This instniment is made in a variety of forms; sometimes 
with a full circle divided into degrees, sometimes comprising 
onfy a semicircle, sometimes npon a rectangular rule having 
not the circumference but the radii drawn, as they would be 
through the divisions of the circumference if it were actuallv 
described. The first kind is made usually of brass, or of 
ver, which is less liable to corrosion, and communicates nc 
pleasant odor to the hands. It has a metallic radius n 




8 PLANE THiaONOH&TRT. 

able about the centre of the circle and extending beyond the 
circum&renee. This prolonged radius serves to point out 
the number of degrees, and is armed with a sharp pin ander 
the outer eztreinity for the purpose of pricking the paper, bo 
that when the instrument is removed a hue may be drawn with 
pencil through this point, and that upon which the centre was 
placed ; wbicit line shall be a radius corresponding to the 
number of degrees at which the instrument was set. 

THE SEUI-CIBcrLAR PROTRACTOB, 




which is the one most commonly «een, is a semi-circle of 
brass, (or other metal,) having the greater part of the interior 
K^t out to render the instrument less heavy. 

The setni-circiiiiifeieQce is divided into degrees by marks 
ms^B iq the pietal, and these are numbered from 0° to 180 
(th^ tl^^lbeI in a semi-circumference) both ways, in order 
that the counting may commence with convenience ai either 
Wd. 

71^ degrees ate also sometimes divwled into half degrees, 
wvi line? of dilfereat length aret employed to mark more dis- 
tjnetly every five and every ten degrees." 

. Th^ ceatre is marked by a notch in the straigtit side of 
the insbrume^t, which side is a diqmeter of the senu-eirclct 



La is te^ed gtadnaiioii' 
1 11iiBin»trai&eiitin>r bemtdeauIoTpipcii'.and ■ large 01 
MOaaait. 



SEMI-CIRCULAR PROTRACTOR. 9 

9. In order to explain the use of the instrument here 
described, suppose it be required to draw at the point a in 
the line ab a line making with ab an angle of 22°. 

Place the protractor so c 

that its centre shall be upon 
the point a, and its straight 

edge or diameter upon the . . 

line AB. Then mark the A B 

paper at the point c against the 22d division of the protractor, 
and a line joining c and a will form with ab the angle re- 
quired. 

10. We are now prepared to construct triangles when 
three parts are given, the angles in degrees and the sides in 
feet, yards, or other linear units. 

In order to show the practical utility of trigonometry at 
the same time that we explain the solution of a triangle, let 
us take the following problem in the calculation of distan- 
ces to inaccessible objects. 

Suppose a fort situated upon an island, and a light-house 
upon the main shore, and let the distance from the light-house 
to the nearest salient of the fort be required. 

Measure a line along the shore 
of any length at pleasure, say 500 
yards, beginning at the light-house* 
Then if two lines be imagined to 
be drawn from the extremities of 
the line just measured, to the sa- 
lient of the fort, a large triangle 
will be formed having its two long- 
est sides resting upon the sea. If 
now the angles which th^se two 
sides form with the first side, which 
we will call the base, could be de- 
termined by observation upon the shore, there would be 
known in this triangle a side and the two adjacent angles, 
which would be sufficient data to construct the triangle on a 
small scale, and to obtain the length of the required side e 
tending from the light-house to the salient of the fort. 

2 





10 PLANE TRIOONOMETRV. 

A somewhat rnde instrument for the purpose of observing 
such angles as those alluded to above, might be easily made. 

Let there be 
a circle, or flat 
circular ring 
of wood gra- 
duated to de- 
grees, and ha- 
ving a tin 
tube moveable 
upon a pivot 
at the centre 
of the circle ; 
the tube being 

closed at one end except a very small orifice, and having 
two threads crossing at right angles in the centre of the other 
end, so that in looking through the tube with the eye at the 
small orifice, the line of sight may coincide with the axis. 
Let this apparatus be mounted upon a three legged stand 
called a tripod, so that the plane of the circle shall be hori- 
zontal : then, by placing the instrument thus formed at the 
light-house, in the example above, and sighting with the 
tube, first to a staff at the other extremity of the base, and 
then to the salient of the fort, keeping the circle stationary, 
the number of degrees passed over upon its circumference by 
the tin tube will indicate the angle of the triangle at the 
light-house. This angle we shall suppose to be 105J^. The 
angle at the other extremity of the base might be found in 
the same manner, and suppose it 47°.* 

To construct the triangle with these data, 
draw on paper a line ab, and make it equal in 
length to five hundred divisions of some scale 
of equal parts.t Then draw an indefinite line 
AC, making with ab an angle of 103^°. Also 
lay off in a similar manner at the point b an 




The inetrument here described is of course very rude. It wis deemed not 
sable to encumber the work with a detailed description of more accurate in- 
tnents, which belongs properly to a treatise on surveying. 
r/r/5 may be done conveniently by taking 60 divisions, and considering each 
iion AB equal to ten. 



SOLUTIONS BT COKBTBUCTION. 



11 



angle of 47^, and the two lines ac and ec will meet at c. 
Take the line ac in the dividers and apply them to the scale. 
The number of equal parts upon the scale between the feet 
of the dividers, will show the number of yards from the light- 
house to the fort. This number is 791. 

If the angle at c were required, it might be measured by 
ap^diying to it the protractor ; or it is equal to 180^ — (A-f b.) 

The side b c if among the sought parts might also be mea- 
sured from the scale. 

11. The instrument described above may be rendered suit- 
able for application to the determination of heights. If a 
round bar be made to project horizontally from the top of the 
tripod, so that the graduated circular frame can be suspended 
by the socket at its centre in a vertical position, it will then 
serve to measure angles in a vertical plane.* 

To show the use of the instrument thus prepared, take the 
following problem. 

Required the height of a tower which stands upon horizon* 
tal ground, and the base of which is accessible. 

Measure 
back a dis- 
tance from 
the base of 




.UW l««U 



the tower, 
say 200 feet; 
call this dis- 
tance the base line ; at the extremity of the base line place 
the instrument arranged for taking vertical angles ; suspend 
a plumb line from the centre of the circle, and the point 90^ 
distant from that in which the plumb line cuts the circum- 
ference will be the point through which a horizontal radius 
would pass. Then sight with the tube to the top of the 
tower ; the number of degrees between the tube and the hori- 
zontal radius just mentioned, will be the measure of the angle 
included between a line drawn to the top of the tower and 
the base line ; let this number be 30^. Constructing a right 



* A vertical plane it one perpendicular to the aorfiice of the earth. 



13 fLANE TRIGONOMETRir. 

angled triangle upon paper, having its ba[^'200 and angle at 
the base 3(P, the perpendicular of thi^' triangle will be the 
height of the tower. The height of the instrument must be 
added to the result found. 

N. B. The sides found will always be expressed in units 
of the same kind as the base. 

12. It is evident that when any three parts of a triangle, 
one of which is a side, are given, the other three may be dis- 
covered by a process similar to those just exhibited. 

This kind of solution is said to be by construction. 

The accuracy of the results must depend upon the nice- 
ness of the instruments, and the care with which the construc- 
tion is made. 

A degree of accuracy so uncertain and so variable, is quite 
inadequate for many purposes to which Trigonometry is 
applied. 

A method oi calculating the required from the given parts 
of a triangle, which should produce always the same results 
from the same data, and be either perfectly, or so nearly ex- 
act, as to leave an error of no importance, however great the 
dimensions employed, would be evidently a desideratum. 
Such a method we have, and it is that which it will be the 
object of the residue of the present treatise to unfold. 

Togive the student a general view of what is before him, 
it will be well to state that a number of equations will be 
found, each containing four quantities, which quantities will 
be general expressions for the measures of parts of a triangle. 
The equation will express the true relation between these parts. 
By making one of these parts the unknown quantity and 
resolving the equation with respect to it, its value will be ex- 
pressed in terms of the other three. If now these three were 
given, the value of the fourth would be known the moment 
the values of the three given were substituted for their gene- 
ral representatives. 

It is plain that as many such general equations will be re- 
quired, as there can l)e formed essentially different combina- 
tions of four o it of the six parts of a triangle. 

Equations like those here alLided to are called formulae be- 



TRiaONOMETAICAL LINK«. 13 

cause each is a general form, under which a multitude of par- 
ticular examples are included. 

As these general forms require of necessity the use of alge- 
braic symbols and processes, and as algebra, from its power 
and application to decompose combinations of quantity so as 
to extricate their elements, is often called analysis, the subject 
upon which we are now about to enter is called 

ANALYTICAL TRIGONOMETRY. 

13. The sides and angles of a triangle are not quantities of 
a similar kind, and therefore do not admit of direct compari- 
son. Since angles are expressed in degrees, and sides in units 
of length, one of the first principles of equations, namely, that 
the members and terms should express quantities of the same 
kind, would be violated by the introduction of angles and 
sides together, without some modification of one or both. 

The expedient which has been invented to accommodate 
these heterogeneous quantities to each other, is that of em- 
ploying straight lines, so related to the arcs which measure 
the angles of ^ triangle, as to depend upon these arcs for 
their length, in such a manner that when the arcs are known, 
these straight lines may be known also; and vice versa. 
The chords of arcs are plainly hues of this description, and 
chords were at one time used for the purpose of which 
we here speak ; but a more convenient kind of lines, of which 
there are three principal sorts termed sines, tangents and 
secants, of an arc or angle calleJ, when spoken of collectively, 
trigonometrical lines, were introduced by the Arabs, and are 
now in general use. These lines being straight and expressed, 
as they will be found to be, in linear dimensions, like the sides 
of a triangle, they may be employed with the latter in equa- 
tions or formulae ; and when, by the resolution of an equation of 
this description, one of these trigonometrical lines is found:* 
terms of one or more sides of the triangle, the angle to wh 
the trigonometrical line belongs may also be supposed t€ 
known. How the former is known from the 
hereafter explained. Let it be taken for gra 



14 PLANK TRlGONpM£TKY* 

the knowledge of a trigonometrical line is equivalent to the 
knowledge of its arc or angle, and vice versa. 

The trigonometrical lines are sometimes called trigonome- 
trical functions of an arc or angle; the reason for which will 
be understood if we first explain the significationof the word 
function as employed in mathematics. 

One quantity is said to be a function of another, when the 
former depends in any way upon the latter for its value. 

It is said to be an increcLsing function when it increases as 
the quantity upon which it depends increases ; and a decreets- 
itig function when it diminishes as the other increases. 

The quantity upon which a function depends is called its 
variable, because this is supposed to change its value at plea- 
sure, the function changing to correspond. 

Now a trigonometrical line depends upon the magnitude of 
its arc for its length ; it is therefore properly termed a function 
of the arc ; and by way of distinction a trigonometrical func- 
tion. 

Of these trigonometrical lines, we now proceed to explain 
the nature and properties. 



THE SINE. 






14. The sine of an arc is a i>erpendicular let fall from one 
extremity of the arc upon the diameter drawn through the 
other extremity. 

Thus the line mp is the 3^ 

sine of the arc am. 

The same line mp is like- 
wise the sine of the arc bm, 
because it is a perpendicular 

let fall from one extre- / _JL — -*^ 

mity M of the arc upon the 
diameter drawn through the 
other B. 

16. Two arcs, which to* 
gether make a semi-circum- 
ference have, it thus appears, the same sine. Two such arcs are 




TRIGONOMETRICAL LINBfi. 16 

called supplements of each other. A semi-cux^le contains 
18(P. The supplement of an arc is therefore what is left after 
taking the arc from 180?. Thus 80^ is the supplement of 
100°. 70° is the supplement of IKP. 

Two arcs which are supplements of each other, have the 
same sine, or, as it is sometimes expressed, the sine of an arc 
is equal to the sine of its supplement. 

If a represent an arc of any number of degrees, the notation 
employed lo express the sine of that arc is sin a. The propo- 
sition* above, stated algebraically, will stand thus, sin a = sin 
(180°— a.) 

The sine of an arc is also the sine of the angle measured 
by that arq. 

16. When the arc is very small, it is plain that its sine will 
be very small also, and that when the arc is 0, the sine will 
beO. 

As the arc increases the 
sine increases till the arc is 
90^, which, being a quarter 
of the circumference, is called 
a quadrant, the sine of which 
is R. (r signifying radius ; 
which line this letter when- 
ever employed hereafter, will 
be understood lo represent.) \ X^ 

As the arc increases be- 
yond 90^, the sine dimi- 
nishes, i. e., becomes a decreasing function of the arc till the 
arc reaches 180°, when the sine is again. Beyond this value 
of the arc the sine again increases till the arc reaches 270®, 




* The word proposition is here used in the enlarged sense of any thing pro- 
pounded as true. 



16 



PLANE TRIGONOMETRY. 




or three quadrants, when the sine is again equal in length to r. 

From 270O to 360° the 
sine decreases till at the lat- 
ter value it is a third time 0. 
Beyond 360° we pursue the 
same round again, and no 
new variations are deve- 
loped. 

17. The least value of 
the sine, is 0. It has this 
value at 0° at 180, and at 
360. 

The greatest value of the sine is r. It has this value at 
90° and at 270. It has all possible values between and r, 
but it has no different values, as the arc increases to two, 
three and four quadrants, from those which it had in the first. 
So that when the sine of an arc greater than 90° is required, 
an arc, having an equal sine, may be found in the first quad- 
rant. To find this arc we have the following rule, the cor- 
rectness of which, the annexed diagram will show. Observe 
how many degrees distant the termination of the given arc 
is from 180° or 360, according to which of these two is 
nearest, and that number of degrees will be the arc in the 
first quadrant, having the same sine as the given arc. 

For example, let the 
given arc be 200°. This 
is nearest 180°, and dif- 
fers 20° The sine of 20° 
is equal in length to the 
sine of 200°. Or m p, 
which is the sine of a b m, 
i3 filso the sine of b m. 

Again, let the given arc 
be 300°, This is nearest 
360, and differs 60. The 
sine of 60° is equal in length to the sine of 300. 

If the given aro exceeds 360°, subtract 360, and then 




TRIGONOMETRICAL LINES. 17 

apply the rule just given. If the arc contains a number of 
circumferences, divide by 360, and apply the rule to the re- 
mainder. 

18. It is customary, for the purpose of bein^ able to bring 
the trigonometrical lines as they appear in the figure, the 
more readily before the mind when the figure is not present, 
to begin all arcs at the same point ; and the point commonly 
chosen is the extreme right of the circumference, determined 
by the intersection of the horizontal diameter of the circle, 
with the circumference. This is the point a, in the last 
figure. An arc of 90^ will then reach to the top of the cir- 
cle, or the upper extremity of a vertical diameter. An arc of 
180° will terminate at the left of the circle, or of the hori- 
zontal diameter. An arc of 270*^, at the lowest point of the 
circle, or lower extremity of the vertical diameter. An arc of 
360^, at the right of the circle, or point of beginning. 

One advantage of this plan will readily appear. Since the 
arc always commences at the same point, namely, the right 
of the circle, the horizontal diameter will be the diameter 
which passes through one extremity of the arc, and wherever 
the arc may terminate, the perpendicular from the other ex- 
tremity of it, which is the definition of the sine, will be a 
perpendicular to the horizontal diameter ; so that the sines 
of all arcs, in a diagram so constructed, will be perpendicu- 
lars to the horizontal diameter. 

The sines of arcs between 0° and 180, will be drawn 
downwards ; and those of arcs between 180° and 360, will 
be drawn upwards. 

According to the general principle of algebra, that quan- 
tities estimated in a contrary sense, are distinguished by con- 
trary signs, if the sines of arcs between 0° and 180, be con- 
sidered as positive, those of arcs between 180° and 360° must 
be regarded as negative.* 



* The student is referred to the well known problem of the Couriers, Art U 
Oaries' Bourdon. 



18 



PLANB TRIGONOMETRY. 




THE TANGENT. 

19. The tangent of an arc, is a perpendicular drawn to 
the radius at one extremity of the arc, and terminated by the 
radius produced, which passes through the other extremity. 

In the annexed 
diagram a t is 
the tangent of the 
arc A M. It is 
also the tangent 
of the angle mea- 
sured by the arc. 
The shorter the 
arc is, the shorter 
will be the tan- 
gent. When the 
arc is 0, the tangent will evidently be 0. ,As the arc increases 
the tangent increases, and very rapidly as the arc approaches 
90°. In order to trace the tangent through its various 
changes, we shall suppose the arc to commence at the point 
on th^ extreme right of the circle, and the degrees to be 
counted upwards, towards the left, as in a former case — the 
tangent of every arc will then be drawn at the extremity of 
the horizontal radius on the right of the centre, and be ter- 
minated byjthe radius produced, passing through the other 
extremity of the arc, which extremity will vary its position 
as the arc varies its magnitude. 

When the arc is 90°, the 
perpendicular to the radius 
at one extremity, is paral- 
lel to the radius through 
the other extremity. These 
lines will never meet, and 
the tangent will have no 
termination. It is in this 
case said to be infinite. 
The sign employed to ex- 
ess infinity oo is also 
lied the sign of impossi- 




TRIGONOMETRICAL LINES. 



19 



bility. The value of the tangent of 9(P, is expressed algebrai- 
cally thus, tan 90° = oo. 

The tangent of an arc, terminating in the second quadrant, 
will be cut off below the origin* of the arc. Thus a t is 
the tangent of a m ; and 
according to the principle 
adopted when treating of 
the sine, this tangent being 
in the opposite direction to 

that of the tangent of an I ^5^ lA 

arc in the first quadrant, 
is nef];ative. 

When the arc is ISO^, 
the negative tangent which 
became shorter and short- 
er, as the second extremity of the arc approached this point, 
again reduces to 0. Beyond 18(P, or in the third quadrant, 
the tangent is cut off above the origin again. Thus a t in 
the annexed diagram, is the Tj 

tangent of the arc a b m. 
The tangent of an arc in 
the third quadrant is, there- 
fore, positive. Wlien the 
arc is 27(P or 3 quadrants, 
the tangent becomes paral- B 
lei to the radius which pro- 
duced ought to terminate 
it, and the tangent is 
again oo. 





* A term applied to the point wliere the arc commences. 




90 PLANE TRIGONOMETRY. 

The tangent of an 
arc in the fourthquad- 
rant is negative, as 
may be seen from the 
annexed diagram. 

20. The Ictast vahie 
of the tangent is 0. 
The greatest vahie is 
oo 4 So that the tan- 
gent has all possible 
values. But these it 
has if we do not re- 
gard the sign, in the first quadrant ; and the same rule applies 
to finding the length of the tangent belonging to any given arc, 
from that of an arc in the first quadrant, as was given for the 
sine. 

The tangent changes its sign in every quadrant, that is 
four times in going round the circle. It is positive in the first 
and third, two diagonal quadrants, and negative in the second 
and fourth, the other two diagonal quadrants. 

The tangent is oo at (he top and bottom of the circle, and 
on the right and left. 

THE SECANT. 

21 . The secant of an arc is a line drawn from the centre 
of the circle to the extremity of the tangent. 

In the preceding dia2:rams, ct is the secant of the arc am 
It is also the secant of the angle measured by the arc. 

As the arc with its tangent diminishes, the secant dimin- 
ishes ; and when the arc and tangent are 0, the secant is equal 
to R. The secant can never be less than radius, because the 
tangent cannot pass within the circumference, and conse- 
quently the line from the centre to the extremity of the tan- 
gent, must extend at least to the circumference. When the 
arc is 90° the secant is oo . When the arc is 180° the secant 

again. And when the arc is 270° or three quadrants, 



TRIGONOMETRICAL LINES. 31 

the secant is again oo . All which will appear from an in- 
spection of the last diagrams. 

The tangent and secant have their greatest values, namely 
00 , together ; that is at the top and bottom of the circle. They 
have also their least values, that of the tangent being 0, and 
that of the secant r, together, to wit, at the right and left 
points of the circle. 

22. In the first quadrant the secant is estimated from the 
centre towards the second extremity of the arc ; in the second 
and third quadrants it is estimated in the opposite direction. 
According to the principle which it is necessary to observe, 
and of which we have before spoken, the secant must in these 
quadrants be considered as negative. In the fourth quadrant 
the secant is again estimated towards the second extremity of 
the arc and is therefore positive. 

The vertical diameter separates the positive from the neg- 
ative secants, the positive being in the quadrants on the right 
of this diameter, and the negative being on the left. 

23. We have now exhibited three of the trigonometrical 
lines. There are three others closely connected with these 
in character, called the cosine, the cotangent and the cose- 
cant ; the reason for which names will presently appear. 

The difference between an arc or angle and 90° or a right 
angle, is called the complement of the arc or angle. Thus 
40° is the complement of 50° ; 60° is the complement of 30° ; 
and in general 90° — a is the complement of the arc a. The 
cosine, cotangent and cosecant, are the sine, tangent, and 
secant of the complement. Thus the cosine of 60° is the 
sine of 40°; the cotangent of 30° is the tangent of 60° ; and 
in general the cosine, cotangent or cosecant of the arc a is 
the sine, tangent or secant of 90° — a. 



23 



PLANE TRIGONOMETRY. 



THE COSINE, 




24. In the annexed 
diagram dm is the 
complement of the arc 
AM ; and Ma being a 
perpendicular from 
one extremity m of 
of the arc dm upon the 
diameter which passes 
ttirough the other ex- 
tremity D, is the sine 
of the arc dm. There- 
fore by the definition 
it is the cosine of the arc am. But Ma = cp. Hence cp is 
also the cosine of the arc am. We have then another defini- 
tion for the cosine of an arc, viz., the distance from the foot 
of the sine of the arc to the centre of the circle* 

25. If the arc terminates on the right of the vertical dia- 
meter, i. e., in the first or fourth quadrant, the foot of the sine 
will fall on the right of the centre ; but if the arc terminates 
on the left of the vertical diameter, i. e., in the 2d or 3d quad- 
rant, the foot of the sine will fall on the left of the centre. 
The cosine being estimated in opposite directions in these 
two cases must have opposite signs. It is therefore positive in 
the 1st and 4th quadrants, and negative in the 2d and 3d. 

It will be recollected that the positive were separated from 
the negative secants, as the positive are here seen to be from 
the negative cosines, by the vertical diameter. The secant 
and cosine have therefore always the same algebraic sign. 

It was shown (art. 15,) that sin (180^ — a) = sin a ; hence 
cos (180° — a) is equal in length to cos a, since they are both 
the distance from the foot of the same sine ( mp in the dia- 
gram of art. 14) to the centre. But if a < 90°, it follows that 
180° — a terminates in the second quadrant, hence its cosine 
is negative ; if a > 90° then cos a is negative, and 90° — a 

ing in the first quadrant, its cosine is positive ; therefore, 



XBiaONOMETRICAL LINKS. S3 

the cosine of an arc and its supplement are equal with the 
contrary sign. 

26. The cosine of 0^ (being equal to the sine of the com- 
plement of (P which is 90°) is r. The cosine of 90° is equal 
to the sine of 0° which is 0. The cosine of 180° being the 
distance from the foot of the sine to the centre, and being also 
on the left of the vertical diameter is — R,asmaybeseen fiom 
the preceding diagram. The cosine of 270° being the dis- 
tance from the foot of the sine to the centre, since (he sine 
fidls on the centre, is 0. 

The least value of the cosine is 0; the greatest value is r. 
When the sine has its least value, the cosine has its greatest; 
and vice versa. 

27. Before noticing the cotangent and cosecant, let us com- 
sider the manner of treating negative arcs. Such arcs com- 
mencing at the point 
A in the diagram 
ought evidently, on 
the general principle 
already repeatedly 
mentioned, to be laid 
off upon the circum- 
ference in the opposite 
direction from the 
positive arcs, i. e., 
downwards. 

Let us for simplicity 
suppose the arc in question to be less than a quadrant; being 
laid off downward, such an arc will terminate in the fourth 
quadrant Hence we see that the trigonometrical lines of a 
negative arc must be afiSscted with the same signs as thor 
an arc in the fourth quadrant. Thus the sine of a nef 
arc will be — , the cosine +, the tangent — ^ the secant 

Secondly, suppose the given negative arc to be g 
than a quadrant ; were it positive, some of il x 

lines would be negative. The rule gives 
Ermines the sines of its trigonometrical 1 
arc in the 4th quadrant will apply with 




24 PLANE TRIGONOMETRY. 

that when the trigonometrical line is + in the fourth quad* 
rant, the corresponding trigonometrical line of the negative 
arc has the same sign as that of a positive arc of the same 
magnitude, and when the trigonometrical line is — in the 
fourth quadrant, a contrary sign. 

The truth of this assertion may be seen, by trying nega- 
tive arcs of various magnitudes upon the diagram, laying 
them off downwards from the right point of the circle, and 
observing in which quadrant their extremities fall. They 
will be found in every case to give results agreeable to the 
rule just stated. 

THE COTANGENT AND COSECANT. 

28. The cotangent of 0° is equal to the tangent of 90° 
(art. 23) and is therefore oo. The cotangent of 90° is equal 
to the tangent of 0° and is 0. The cotangent of 180° is 
equal to the tangent of 90° — 180° = the tangent of— 90° 
■» 00 since — 90° is a negative arc, and terminates at the 
bottom of the circle, or the 270° point. The tangent of 270° 
« the tangent of 90° ~ 270° = the tangent of — 180° = 0. 

When the tangent has its least value which is 0, the co- 
tangent has its greatest which is oo, and vice versa. 

29. The cosecant of 0° = the secant of 90° = oo. The 
cosecant of 90° == the secant of 0° = r. The cosecant of 
180° = the secant of 90° — 180° = oo. The cosecant of 
270° = the secant of— 180° = 0. 

When the secant has its least value which is r, the cose- 
cant has its greatest, which is oo, and vice versa. The co- 
tangent and cosecant have their greatest values together and 
their least values together, viz., that of the one 0, of the other 
R, at the top and bottom of the circle, and both oo at the right 
and left points. 

30. With regard to the signs of the cotangent and cosecant 
i the different quadrants, they will be most conveniently 

liscovered from the analytical expressions for these lines 
which we shall presently have. We add here, however, 



TRlQONOMETRtCAL LINfiB. 26 

which SO far as the cotangent and cosecant are concerned 
must be for a moment taken for granted, that the six trigono- 
metrical lines may be arranged in three pairs, each pair 
having always the same algebraic sign. 

We have seen that the secant and cosine go together in 
this way ; so do also the cosecant tmd sine ; and so do the 
tangent and cotangent. The positive sines and cosecants are 
separatexl from the negative, by the horizontal diameter ; the 
positive cosines and secants from the negative, by the vertical 
diameter ; and the tangent and cotangent are together -|- and 
— alternately in the successive quadrants. 

31. The following algebraic notation is employed for the 
six trigonometrical lines. Let a be the algebraic expression 
for the number of degrees in any arc, then the trigonometri- 
cal lines of the arc a will be expressed thus; sin a, tan a, 
sec a, cos a, cot a, cosec a. 

Cot a tan a = r is read, the cotangent of the arc a mul- 
tiplied by the tangent of the same arc is equal to the square 
of the radius of the circle in which these trigonometrical 
lines are supposed to be drawn. Cot a and tan a are expres- 
sions for straight lines, and the equation above expresses that 
the rectangle formed by the tangent and cotangent of an arc 
is equivalent to the square formed upon the radius. 

The two members of the above equation contain the same 
number of dimensions, and are therefore homogeneous. This 
ought to be the case in all trigonometrical equations ; because 
a line cannot be equal to a surface, nor either of these to a 
solid. 

Sometimes in analytical investigations r is supposed to be 
equal to 1 ; r' and r^ would also be equal to 1. Whether 
this 1 is a unit of length, of surface, or of solidity, must be 
determined by what is required to preserve the homogeneity 
of the equation. 

32. The tangent, secant, cotangent and cosecant may be 
expressed in terms of the sine and cosine. 



26 



PLANE TRIGONOMETRY. 



The values of the four former in terms ef the two latter are 
derived geometrically as follows : 

Call the arc am 
in the diagram a, 
then DM « 90° — 
a = complement 
of a, DE = cot a 
and CE = cosec a. 

In the similar 
triangles CPM and 
CAT, since homo- 
logous sides are 
proportional we 
have 




or 



cp : PM : : CA : AT 



cos a : sin a : : R : tan a 



whence multiplying the means and dividing by the first term, 

we obtain the last 

R X sin a 



tana = 



cos a 

that is the tangent of any arc is equal to radius multiplied 
by the sine divided by the cosine of the safne arc. If r 
be made equal to 1^' then 

sin 



tan = 



cos 



33. In the same similar triangles we have 

CP : CM : : CA : ct 



or. 



cos a : R : : R : sec a 



hence, 



sec a = 



when R = 1 



sec = 



cos a 
cos 



/ 



TRIGONOMETRICAL LINES. ' 27 

34. In the triangles cmp and ced, which have their sides 
respectively parallel, and are therefore fdliiilar, we have the 
proportion* 

MP : CD : : cp : DE 



or, 



whence, 



sin a : R : : cos a : cot a 
R cos a 



cot a = 



when R = 1 



sin a 
cos 



cot = 

sin 



36. The same triangles give also the proportion 

MP : DC : : CM : ce 



or. 



whence, 



R being 1 



sm a : R : : R : cosec a 



R' 



cosec a = 



sin a 



\ 

cosec =- -^- 
sin 



36. In the expressions for the tangent and cotangent which 
we have here derived^ it will be observed that we have the 
quotient of the sine and cosine, and that therefore when the 
sine and cosine have contrary signs, the tangent and cotangent 
will be negative. This occurs in the second and fourth 
quadrants. 

It appears hence, that the cotangent changes its sign always^ 
with the tangent. 

Also that both tlie tangent and cotangent of an arc are 
equal to those of its supplement with contrary signs. 

From the expressions for the secant and cosecant, it ap^ 
pears that the former must always have the same sign as 
the cosine, and the latter the same as the sine. 

The formulae derived in the last four articles should be 
committed to memory. 



* The homologous sides are Ihese which are parallel. (Geom. B^ 4, Prop. 21, Schol.) 



28 



PLANE TRIGONOMETRY. 



37. Multiplying the expression for the tangent given in 
art. 32 by that of the cotangent in art. 34, we have 

tan a cot a = r* 
whence, 



tsoi =1 



R' 

cot 



and, 



R^ 

cot =: 7- 

tan 



38. We are now prepared to find formulee for the solution 
of right angled plane triangles in all cases, and plane triangles 
in general in a few particular ones. The remaining cases of 
triangles in general will require further preliminary matter. 

DERIVATION OF FORMULAE FOR THE SOLUTION OF RIGHT 

ANGLED PLANE TRIANGLES. 



Let ABC be any 
right angled triangle. 
With c as a centre 
describe, with any ra- 
dius at pleasure, the 
arc MN terminating at 
the sides of the angle. 
This arc will be the C 
measure of the angle c. 




M 

I \ 
I \ 



.1. 



P N 

Draw MP perpendicular to cn. mp 
will be the sine of the arc mn because it is drawn from one 
extremity m of the arc perpendicular to the diameter which 
passes through the other extremity n. 

MP is also the sine of the angle c. cm is the radius of the 
circle to which the arc mn belongs. 

The two triangles cmp and cba are equiangular and simi- 
lar, and give the proportion. 

CM : MP : : CB : BA or R : sin c : : cb : ba. 
Had an arc been described with b as a centre in a similar 
manner we should have had r : sin b : : bc : ba from 



RIGHT ANGLED TRIANGLES. 29 

which it appears that the radius of any circle whatever^ 
bears the same proportion to the sine in that circle of the. 
arc which measures one of the acute angles of a right angled 
triangle^ that the hypothenuse of the triangle does to the side 
opposite the acute angle. 

It is customary, for conciseness, to represent the sides oppo- 
site the angles of a triangle by small letters of the same name 
with the large letters which are placed at the angles ; which 
large letters are also employed as the algebraic representa- 
tives of the angles. Thus in the triangle above, a being the 
right angle, the hypothenuse opposite is expressed by a ; the 
side AC opposite b is represented by 6, and so for the other. 
The above proportions would, according to this method, be 
written thus 

R : sin B : : a : 6 
R : sin c : : a : c 

Both these proportions are expressed in the single rule 
printed in italics above.t 

The two acute angles of a right angled triangle are toge- 
ther equal lo a right angle or 90°. (Geom. B. 1, Prop. 26, 
Cor. 4,) therefore they are complements of each other ; hence 
sin c = cos B ; and the second of the above proportions may 
be changed into 

R : cos B : a : c 

which may be translated into ordinary language thus; radius : 



* It i« important to observe that the same trigonometrical lines of angles or 
arcs containing the same number of degrees in two different circles bear the 
tame relation to each other. Thus in the diagram above, cm : mp : cm: up 
or, 

(a : sin) of the smaller circ : : (r : sin) of the larger 
also, 

MP : cp : : MC : cp or (r ; cos) of the one : : (r : cos) of the other, 
t When R = 1, multiplying the second and third terms and dividing hy the 
first, in the preceding proportions we have 

6 = a sin B 
and, 

c =ia sin c 
That is tUker side = tht hypoth. x the stTUxof the angle opposU 



80 PLANE TRIGONOMETRY. 

the cosine of one of the acute angles of a right angled trian- 
gle : : the hypothenuse : the side adjacent the acute angle.* 

When any three terms of a proportion are given, the 
remaining term can be found. If the unknown term be one 
of the extremes, multiply the two means and divide by the 
other extreme; if the required t^rm be a mean, multiply the 
two extremes and divide by the other mean. 

The above proportions contain each of them two of the 
sides of a triangle, the sine or cosine of an angle, and radius. 
If the lengths of the sides be given in numbers, these numbers 
may be put in place of the small letters which represent the 
sides in the proportion, and the general form becomes so far 
adapted to a particular casein the solution of right angled 
triangles ; but if the angle is given in degrees, how are we to 
know its sine or cosine, for that is the quantity which enters 
into the proportion ; and how are we to know the numerical 
value of R? For the present the student must be satisfied 
with the reply, that he can find the numerical value of any 
trigonometrical line corresponding to an angle of any given 
number of degrees, in a table at the end af the work. This 
is Table III, entitled, A Table of Natural Sines, <fcc. The 
degrees for angles or arcs of every magnitude within the 
quadrant will be found at the top of the columns of the table, 
and the minutes in the first column on the left, if the given 
angle or ar6 be less than 45° ; and the degrees will be found 
at the bottom of the page and the minutes on the right, if it be 
greater than 45°, the length of the sine or cosine will be found 
in the column under or over the degrees, as the case may be, 
and on the same horizontal line with the minutes. The title 
of the column must be looked for at top if the arc be less than 
45°, and at bottom if the arc be greater. The other trigono- 
metrical lines may be easily calculated from the sines and 
cosines, as will be seen in the examples. 



♦ When R = I 

c = a cos B 
either side r= hypotk^ X cos adjac. angle. 



RIGHT ANGLED TRIANGLES. 31 

The trigonometrical lines of this table are computed, by a 
rule which will be hereafter demonstrated, for a circle whose 
radius is 1. So far as the principles for the solution of tri- 
angles areconcemed, the length of the radius is entirely imma- 
terial, as it will be recollected that the arc in the diagram 
above was described with any radius at pleasure. 

When in cases of the solution of right angled triangles, the 
hypothenuse and one of the acute angles are either given or 
required by the problem, one of the above formulae* is alwajrs * 
employed. 

39. Let us take an example by which to illustrate their ap- 
plication, and as upon a former occasion, one which shall at 
the same time exhibit the practical utility of Trigonometry. 

A roof is to have a height 
of 15 feet in the interior at the 
centre, and an inclination of 
35°. 

Required the length of the 
inner line of the rafters. 

A right angled triangle will 
be formed in which the angle 
at the base will be 35^, and the side opposite 15 feet, and of 
which the hypothenuse is required. The first formula of the 
last article applied to this case givest 

1 : sin 35° : : a : 15 

Multiplying the extremes and dividing by the first mean, 
the value of the ether mean which is a, the hypothenuse re- 
quired, will be obtained 

1x15 




a 



sin 35° 



♦ They are under the form of a proportion, but may be converted into equa- 
tions. This remark is here made because we have spoken of equations only as 
being called formulae. Proportions are no less so. 

t Since in the demonstration of the formulae, the sides and angle of the tri- 
angle were supposed to have no particular values, it follows that any numbers^ 
or any other letters compatible with the properties of a triangle, may be put in 
the place of those employed, and the formulae will still be true. This must be 
borne in mind throughout the work. 



32 PLANE TRIGONOMETRY. 

Looking out the sine of 35^ in the tables and performing 
the operations indicated in the last equation, the value of a 
will be known which will be the length of the rafters required. 
The answer will be in feet. 

Sin 35^ is found from the tables to be .57358, hence, 

a = = 26.1 feet 

.57358 — 

If (to vary the problem) half the interior breadth of the 
roof had been given, say 20 feet, and the angle of inclination 
instead of 35^ as in the last example had been 15^, then to 
find the length of the rafters, it would be necessary to find 
first the angle opposite the given side 20 feet ; which is done 
by subtracting the given angle 15° from 90°, since the two 
wute angles are complements of each other. (Geom. B. 1, 
Prop. 25, Cor. 4.) The remainder is 75°. 

Applying the same formula as before, there results the pro- 
portion 

1 : sin 75° : : a : 20 

whence, 

_ 1x20 
^ ~ sin 76° 
The same result might be obtained by using the given 
angle 15°, and employing the last of the general formulae 
above which contains the cosine of on^ of the acute angles. 
The proportion would stand thus 

1 : cos 15° : : a : 20 
whence^ 

1x20 

COS 15 

Finding cosine of 15° in the tables and substituting it here, 
the value of a would be found to be the same as in the last 
result. In fact cos 15° = sin 75°. 

40. Had the height and half the breadth of the interior of 
the roof been given, the length of the rafters might have been 
obtained, by employing the property of the right angled tri- 
angle demonstrated at Prop. 11, B. 4, of the Geometiy, that 

e square on the hypc^henuse is equivalent to the sum of 



RIGHT ANGLED TRIANGLE!. S3 

the squares upon the other two sides. Let the height of the 
roof be 12 feet, and the semi-breadth 16 feet, then 

a" = 12' + 10' = 400 
whence, 

a = 20 

If the length of the rafters had been given equal to 20 feet, 
and the height of the roof equal to 12 feet, then the semi- 
breadth would have been expressed thus 

fe' = 20^ — 12' = 256 
whence, 

6 = 16 

41. Had the semi-breadth or base of the triangle and the 
inclination of the roof been given, and the height of the roof 
or perpendicular of the triangle been required, the hypolhe- 
nuse not entering into the problem, neither of the above for- 
mulae, all of which contain the hypothenuse, would serve to 
find the side required in a direct manner. It might, however, 
be found indirectly by first finding the hypothenuse, using 
one of the above proportions, and then by means of the hy- 
pothenuse, using the same proportion, the required side might 
be obtained. 

It is, however, objectionable to find one of the required parts 
in terms of a part which has itself been calculated from the 
given parts ; because in the use of the tables which give the 
trigonometrical lines of the different angles not with perfect 
accuracy, but truly for as many decimal places as the table 
employs, a small error arises from the decimals neglected be- 
yond the last place, and this though so small as to be unim- 
portant, becomes magnified by repetition, as in the case where 
one part of a triangle itself not perfectly accurate, is employed 
to calculate another. It is therefore desirable to find each of 
the required parts, in terms of the given parts ; and this may 
always be done in right angled triangles. We proceed, there- 
fore, to demonstrate a formula for the direct solution of the 
last case supposed above. 



6 



34 



PLANS TRIGONOMETAT. 



Let ABC be a right 
angled triangle. With 
any radius at pleasure 
describe an arc ml 
which shall be the mea- 
sure of the angle c. At 
the point l draw a per- 
pendicular LT to the line 
CL, terminating at the line ct. lt is evidently the tangent 
of the arc ml, since it is a perpendicular to the radius at one 
jBxtremity of the arc, and is terminated by the radius which 
passes through the other extremity. It is also the tangent of 
the angle c. 

The equiangular and similar triangles clt and cab give 
the proportion 




GL : LT : : CA : AB 



or, 



R : tan c : : b : c 

Let the angle of the roof in the above problem be 20° and 

the semi-breadth 25 feet, then 

1 : tan 20° : : 25 : c 
whence, 

tan 20° X 25 
.c= 

' Had the angle b been used instead of c, the resulting pro- 
portion would have been , 

R : tan b ; : c ; b 
Both proportions may be expressed together in common 
language thus : Radius : the tangent of one of the acute an- 
gles of a right angled triangle : ; the side adjacent that 
angle : the side opposite. 

This last rule applied to the problem at art, 11, gives 

1 : tan 30° • : 200 ; c 
whence,* 

c = tan 30° x 200 = .57735 x 200 = 115.47000 
c is the height of the tower. 



* The tangent is found from Table IIL by dividing the sine by the cosine. 
\rt 32.) Should the cotangent be required, divide the cosine by the sine, (Art 

;.) To find the secant divide 1 by the cosine. (Art 33.) For the cosecant, 
ivide 1 by the sine. (Art 35.) 



THEORY OF LOGARITHM*. 35 

If the side c were given and the angle b, the side b might 
be found in the same manner, nsing the proportion which 
contains the angle b. 

42. We have now exhibited all the cases which can possi-* 
bly occur in the solution of right angled triangles, with some 
specimens of their application. The right angle of the tri- 
angle is fixed ; and any two of the five remaining parts being 
given, the other three may be found. Let the student select 
at pleasure any two of the five parts, the two selected to be 
considered as given, and he will find the case for solution 
with which he will then be presented, included in the exan> 
pies above. 

The operations in the cases already exhibited, though of 
the most simple kind, nevertheless involve multiplications^ 
which, from the number of places of figures, are somewhat 
tedious. In more complicated cases this evil would be much 
increased* 

On this account it is customary to employ in trigonometri- 
cal calculations, that ingenious invention of Lord Napier's for 
facilitating numerical calculations, the table of logarithms ;* 
before explaining the use of which we shall give a tolerably 
full exposition of the 

THEORY OP IX)GARITHMS* 

43. The logarithm of any given number is the exponent of 
the power to which it is necessary to raise some particular 
number in order to produce the given number. Thus^ let 
10 be the number raised to the power ; then 2 is the loga- 
rithm of 100, because 10^ = 100 and 3 is the logarithm of 
1000, because 10' = 1000. Every given number will have 
a corresponding logarithm or exponent of the power to which 
it is necessary to raise 10 in order to produce the given 
number. 

The number 10, which is the only number that does not 
change in the above equalities, is called a constant. 



*Tabl«Lattho«Bd^ 



86 PLAN£ TRIGONOMETRY. 

Should the constant number which has been employed be 
changed for another, the logarithms of numbers would be 
different from those derived by the use of the first constant. 
Logarithms derived from different constants are said to belong 
to different systems of logarithms, and the constant number 
belonging to each system is called the base of that system. 
The system most in use has the number 10 for abase, and is 
called the common system. The relation which this number 
sustains to the decimal system of notation will readily 
suggest some reasons for its selection ; it will be found, as we 
proceed, to have many advantages. 

44. If 6 be the base of a system, n a number, and I its loga- 
rithm, then by the definition 

If we put b in the place of w, this equation becomes 

Here I is evidently equal to 1. Hence the logarithm of 
the base of every syste7)i is 1. 
If in the equation 

b^ = n 
we make n = I we have 

b'=l 
Here lis evidently equal to zero. (Davies' Bourdon, art. 52.) 
Hence in every system log. o/' 1 = 0. 

45. Suppose now the system be the common system ; b 
will be equal to 10. If we substitute for n all possible num- 
bers successively, we shall have a series of equations like the 
following, 

10^=1 

W = 2 

10' = 3 
&c. 
In the first I is the common logarithm of 1, in the second of 
2, in the third of 3, &c. If I be made the unknown quantity, 
and these equations be successively resolved, we sliall have 
the common logarithms of all numbers.* If now a table be 



♦ The method of resolving them is given at Art. 238, Alg. Davies* Bourdon. 



THEORY dF LOGARITHMS. 37 

formed having the series of natural numbers 1, 2, 3, 4, &c. in 
one column, and their logarithms calculated as above placed 
in a second column against them, this would be a table of 
logarithms. The tables in actual use do not ditfer from such 
an one in principle, though some arrangements are adopted in 
them to avoid unnecessary repetitions. 

46. A better method of calculating tables of logarithms, is 
by means of series. A series for the purpose will be found 
at the top of the last page of Art. 253 in the Algebra. It is 
as follows, except that /', which is the value that I takes when 
A of art. 249, is made equal to 1, has been changed back to I 
and A has been replaced, 

/(z+l)-fo=A2(-^+3-^^3+g^-2-J:pjp+&c.) 

Making in this z =1 we have 

But /I == (art. 44.) hence the second member above is the 
logarithm of 2. 

This method does not give the logarithm with perfect 
accuracy, since it is impossible to employ all the terms of the 
series which extends to infinity. But it will be observed that 
the terms, as we advance in the series, become very small 
fractions, and will soon be too insignificant to make any 
material difference in the sum of the whole. 

Calculations which furnish very nearly, but not exactly, 
the true value of quantities, are called approximations. The 
resolution of the exponential equation is by approximation, 
and the logarithms of numbers in general, can be found in 
no other way.* 

Again making 2; = 2, the first member becomes Z3 — 12. 
But 12 has been just found, and may be substituted ; and being 
transposed to the second member, the whole second rnemb 



♦ For the method which was actually employed in calculating the first t 
see the article entitled "Invention des Logarithmes," in iheio^" * "^ 
tables of Caliet. Also article Logarithms, in the Endinburgh ' 



38 PLANE TRIGONOMETRY. 

will then be the logarithm of 3. Making again z = 3 we 
should have log. of 4 in terms of log. 3 (which has just been 
found) and a series. Thus we might proceed to find the loga- 
rithms of all numbers.* It will be perceived that the letter a 
remainis in all the results, as indeed it ought ; for we are sup- 
posing no particular system of logarithms, and the value which 
is given to a, will determine the system to which the above re- 
sults shall refer. The calculation would be most simple, and 
the labor of making a table of logarithms least, on the supposi- 
tion of A = 1. This is the value of a, which belongs to the 
Naperian system, or that of Napier, the author of the first 
tables. If A be any number besides 1, then a different system 
from the Naperian will be furnished by the series, and the 
new value of a will be the quantity by which ft is necessary 
to multiply the Naperian logarithm of a number to have its 
logarithm in the new system. This quantity iscalled the modu- 
lus of the new system. The modulus is constant for the same 
system. If a table of logarithms be required with a given 
modulus, it may be formed from a table of Naperian loga- 
rithms, by multiplying each logarithm in the Naperian table, 
by the given modulus. 

We shall show how the modulus may be found far the 
common system, and this will point out the method for every 
other. The method would in fact, be in all respects the same, 
changing com. log. in the following investigation, into the 
lograrithm of the system under consideration. 

By the definition, if a be the modulus, and n any number^ 
then 

a X nap. log. n =*= com. log. n. 

for n substitute the Naperian base, then 

a X nap. log. of nap. base =±: com. log. of nap. base. 
but nap. log. nap. base = l. Substituting this value in the 
first member of the last expression, it becomes 

A = com. log. of nap. base. 
From which it appears that the modulus of the common 
system, is equal to the common logarithm of the base of the 

* The constructing of tables by means of a formula in this wa^, is called 
tabulating the formula. 



THEORY OF LOGARITHMS. 39 

Naperian system. And the modulus of any system is the 
logarithm of the base of the Naperian system, taken in that 
system of which it is the modulus. 

47. We shall confine our attention now to the common 
system, the base of which is 10. 

The logarithm of 10 is 1, the logarithm of 100 is 2 ; and 
the logarithms of all numbers between 10 and 100 are be- 
tween 1 and 2, that is, they are 1 and a fraction. The loga- 
rithm of 1000 is 3, and the logarithms of all numbers between 
100 and 1000 are between 2 and 3, that is, they are 2 and a 
fraction. In the same manner it may be shown that the 
logarithms of all numbers between 1000 and 10,000 are 3 
and a fraction ; of all numbers between 10,000 and 100,000, 4 
and a fraction, and so on. The logarithms of most numbers, 
therefore, are mixed numbers. The fractional part is written 
in the tables ; the whole number part, which is called the 
characteristic, is not written, nor is it necessary that it should 
be ; for numbers between 10 and 100, or those composed of 
two figures, have 1 for a characteristic, as has just been shown ; 
numbers between 100 and 1000, or those containing three 
figures, have 2 for a characteristic ; numbers containing four 
figures have 3 for a characteristic, and so on. Whence it 
appears that the characteristic is always 1 less than the 
number of dibits in the number to which the logarithm be- 
longs. So that if against any given number, the decimal 
part of its logarithm be found in the tables, the entire part or 
characteristic may be supplied by counting the figures in the 
given number, and making the characteristic, one less. 

In proceeding to explain the tables, we will premise that 
the logarithms of several consecutive numbers, if the num- 
bers be somewhat large, will difi*er so little as to have several, 
of their first figures the same. Hence, by a proper arrange- 
ment of the tables, the first figures of the logarithm may be 
written but once for several numbers, provided all be desig- 
nated to which they refer, and thus much repetition be 
avoided. 

The manner in which this is accomplished, will be she 
in the 



40 PLANE TRIGONOMETRT. 

EXPLANATION OF THE TABLES. 
PROBLEM I. 

48. To find from the tables the logarithm of any given 
number. 

Case. I. When the number is less than 100, find it in the 
small table on the first page of the tables, in one of the colunins 
entitled n. The number found on its right upon the same 
horizontal line in the next column marked Log. will be its 
logarithm. 

Case. II. When the number is bettveen 100 and 10,000, 
if it be composed of three figures, find it in the table com- 
, mencing on page 2, and in the column entitled n ; in the next 
column marked at top and on the same horizontal line you 
will find the decimal part of the logarithm required. Tliis 
contains six places, the first two of which being the same for 
seyeral numbers are not repeated, but must be understood 
before those which follow the number that has them expressed 
until you come again to six places. If the given number 
contains four figures, find the first three of it in the column 
N as before, and the fourth in one of the columns marked 0, 
1, 2, 3, &c., at top; under the latter, and on the same hori- 
zontal line with the first three, you will find four places of 
the decimal part of the logarithm sought ; the first two places 
to be prefixed to these, are to be taken from the column 
marked at top^ It sometimes happens that in the columns 
which follow that marked 0, the partial logarithms to which 
the projecting figures in the column refer are exhausted 
before you come to the right of the table. This will be indi- 
cated by the four places in the column where it occurs, ex- 
^pressing a less number than those in the preceding columns. 
The two figures to prefix found in the column are then 
those below instead of above the horizontal line in which you 
iwe. In fiict some of the first places of the four in the last 
column, to which the projecting figures above refer, will in 
this case be ciphers, and these ciphers are represented by dots 
to call the attention ; so that in passing back from the column 
hich has your fourth figure at top, if you pass over dots on 



TABLES OF LOGARITHMS. 41 

the same horizontal line take the two projecting figures below 
in the column marked instead of those above. 

N. B. The characteristic is always one less than the num- 
ber of figures in the given number. 

EXAMPLE 1. 

Required the logarithm of 217. 

In the column entitled n on page 3 of the Tables, I find 217; 
in the next column marked at top, and on the same hori- 
zontal line I find 6460 ; projecting to the left a little above I 
find 33, which I prefix to the 6460 and have 336460 for the 
decimal part of the logarithm required. The characteristic 
is 2, and the whole logarithm 2.336460. 

EXAMPLE 2. 

Required the logarithm of 1122. 

On page 2 and in the column n, I find 112 ; in the column 
having the last figure 2 of the given number at top, and cm 
the same horizontal line with the 112 before found, I find 
9993; projecting to the left in the column and a little above 
the horizontal line in which I am, I find 04 which prefixed 
gives me 049993 or with the proper characteristic 3.049993, 

EXAMPLE 3. 

Required the logarithm of 2188. 

In the column n, I find 218, on the same horizontal line 
with which and under 8, 1 find 0047, the dots being ciphers ; 
I pass to the column 0, but instead of taking the projecting 
figures above, I take those below, namely 34, and I have 340047 
or with the characteristic 3.340047 the logarithm required. 

In the same manner the logarithm of 1178 is found to be 
3.071145. 

49. We proceed now to show the use of logarithms in 
numerical calculations. 



43 PLANE TRIGONOMETRY. 



MULTIPLICATION. 

Let a be the base of the system of logarithms, n any num- 
ber, and I its logarithm. Then by the definition, 

a' = w 

Let w' be another number, and I its logarithm, we have 
also 

Multiplying these two equations, member by member, and 
observing the rule for exponents in multiplication, which is 
to add them together, we have 

From this last expression, it appears that l-\-V is the ex- 
ponent of the power to which it is necessary to raise the base 
a, in order to produce the number nri. But nn' is the pro- 
duct of n and 7i'. Hence the logarithm of the product is equal 
to the sum of the logarithms of the multiplier and multi- 
plicand. 

EXAMPLE. 

Multiply 2421 by 1613. 

The logarithm of 2421 is 3.383995 

The logarithm of 1613 is 3.207634 

The logarithm of 2421x1613 or 3905073 is* 6.691629, or 
the sum of the logarithms. 

If in addition to the numbers n and ri above, we suppose 
a third number n" of which the logarithm is X' we shall have 
in a similar manner 

a '+^'-f'" = wn'n" 
and so on. 

Or in general the logarithm of a product of several factors 
is equal to the sum of the logarithms of those factors se- 
jterately* 



* As the number 3905073 is too large to be found in the tables, the method of 
toding its logarithm from the tables must be postponed to the explanation of such 
ises, in advance. 



TABLES OF LOGARITHMS. 43 



DIVISION. 

Dividing the equation : 
by the equation 



a'= » 



Or = n 



We have, observing, the rule of division, to subtract the 
exponent of the divisor from that of the dividend in order to 
obtain that of the quotient. 

n 



n' 



Since I — f is the exponent of the power to which it is 
necessary to raise a the base, in order to produce -^ it follows 
that I — I' is the logarithm of—; i. e. the logarithms of the 

TV 

quotient is equal to the difference between the logarithms of 
the divisor and dividend, 

EXAMPLE. 

Divide 3906073 by 2421 

The logarithm of 3905073 is 6,591629 

« " " 2421 is 3.383995 

The logarithm of 3905073 _,^ . ^„_.^. ,.^ ., 

0401 — = ^^^^ ^ 3.207634 diff. of logs. 

Before explaining other operations by means of logarithms, 
we shall exhibit some principles derived from those just de- 
monstrated. 

50. The base of the common system being 10, the common 
logarithm of 10 is 1. (art. 44.) Hence if any number be 
multiplied or divided by any number of times 10, the loga- 
rithm of the result will be equal to the logarithm of the given 
number increased or diminished by the same number of times 
1. This 1 being an entire number, the decimal part of the 
logarithm of the given number will not be altered by this 
addition or diminution, but only the characteristic. 



44 PLANE TRIGONOMETRY. 

Thus 397940 '^hich is the decimal part of the logarithm 
of 2500, is also of 25000, and of 250000, or of 250, or of 25. 

The characteristics belonging to these different numbers 
are different. That of the log. of 2500 is 3 ; that of the 
log. of 25000 is 4 ; that of the log. of 25 is 1. 

Any number is divided by a multiple of 10, by pointing 
off from the right as many places for decimals, as the divisor 
is times 10. 

Thus 2348 divided by 10, by twice 10, by three times 10, 
becomes successively 234.8 23.48 2.348. The decimal part 
of the logarithms of these last three numbers, will be the 
same, the characteristic being one less each time that we 
divide by 10 or remove the decimal point one place to the 
left. The characteristic of the first, which is between 100= 
10^ and 1000=103,is 2. The characteristic of the second is 1 ; 
and the characteristic of the last is 0, since 2.348 is less 
than 10, or 10\ 

The decimal part of the logarithm of a number consisting 
of significant figures, either followed or preceded by ciphers, 
will be the same as if the ciphers were absent. Thus the 
decimal part of the logarithm of 482000 or of .00482 is the 
same as the decimal part of the logarithm of 482. 

The following table illustrates the theory of the charac- 
teristic. 

The characteristic of the log. of 482000 is 5 

of 482 is 2 
of 4.82 is 

of .482 is - 1 

of .0482 is - 2 

of .00482 is - 3 

From the above, it appears that the characteristic of the 
logarithm of a decimal fraction is negative ; the decimal part 
of the same logarithm is, however, positive. The actual 
value of the whole logarithm will be therefore a negative 
quantity somewhat less than the characteristic. That the 
logarithms of proper fractions ought to be negative, appears 
from the fact, that since a fraction expresses the quotient of 
le numerator divided by the denominator, applying the rule 



Theory of logarithms^ 46 

for division by logarithms, the greater logarithm would have 
to be subtracted from the lesser and the remainder would of 
course be negative. 
From the above principles are derived the following rules : 

1. To find the logarithm of a number consisting of signi- 
ficant figures with any number of ciphers annexed^ find the 
logarithm of the significant figures ^ and make the cha- 
racteristic one less than the number of figures in the given 
number including the ciphers. 

2. To find the logarithm of a decimal or mixed number, 
consider the number as entire ; find the decimal fart of its 
logarithm, and make the characteristic one less than the 
number of figures in the entire part of the given number. 

4. To find the logarithm of a decimal number having 
ciphers at the left ; look for the logarithm, of the signy 
ficant figures, and make the characteristic negative* and 
one more than the number of ciphers at the left of the 
given decimal, 

EXAMPLES. 

The logarithm of 3226000 is 6.514016 

of 114.1 is 2.057286 

of .001684 is 3.226342 

51. We proceed now to the method of determining the 
logarithm of a number beyond the limits of the table. This 
method is by a simple calculation from the logarithms of 
numbers which the table contains, and depends upon the fact 
that the difference of any two numbers bears the same pro- 
portion to the difierence of their logarithms, that the differ- 
ence of two other numbers does to the difference of their 
logarithms, which is nearly true. 

Take two numbers in the table differing from each other 
by 100 as the numbers 843700 and 843800 and a third num- 
ber 843742 differing from the first of these by 42. The log? 



♦It is customary to write the negative sign over the characteristic,thus, 2.171 
It affects the characteristic alone and not the decimal part of the rosflurithtn* 

8 



46 PLANE TRIGONOMETRT^ 

rithm of the first number 843700 is given by the tables and 

is 5.926188 
The logarithm of the second number 843800 is 5.926240 



Thier difference is •52 

which may be found by subtraction, but to save this trouble 
the subtraction is performed and the difference is written in 
the column marked d, the last of each page in the table. 

Then 

diff. of numbers difToflogs. diff. ofnum. diff. oflogsu 

100 : .000051 :: 42 : x 
hence ^ _ mmix^A2 _ ^^^^^^ 

adding this to the logarithm of 843700 which is 5.926188 

.00002142 



the sum, rejecting the last two places 42 which go 

beyond the usual number is 5.926209 

which is the logarithm of 843742. 

Had the first two numbers differed by 1000 instead of 100 
the divisor in the value of x would have been 1000 and the 
quotient would have extended three places beyond the usual. 
The inaccuracy of this method increases with the number 
of^ additional figures beyond four, in the number the logarithm 
of which is to be found. 
From the above process may be observed the following rule: 
To find the logarithm of a number beyond the limits of the 
table. Enter the table with the first four figures of the given 
number, and find the corresponding logarithm. From the 
column marked d take out the number opposite to this lo«[a- 
rithm, and multiply it by the remaining figures ef the pro- 
posed number, reject from the product as many figures to the 
right as there are in the multiplier, and add the rest of th« 
product to the logarithm already found. 



* The remainder is 52, but if the decimals had been carried beyond six places 
'n the tables, it would have been 51. 



TABLES OP LOGARITHMS. AT 

EXAMPLE I. 

Required the logarithm of 739245. 

The decimal part of the log. of 7392 is 868762. 

the number in column d is 69 

Multiplying this by the remaining figures of the 

given number 45 

"^ 
236 



Product, 2665 
From this product reject as many figures to the 
right as are contained in the multiplier, that is two 
in this case, and add the rest to the logarithm before 
found, namely 868762 

The sum is 868789* 

which is the decimal part of the log. of 739245 required* 
Prefixing the proper characteristic, we have 5.868789. 

EXAMPLE II. 

X 

Required the log. of 8193217 

log. of 8193 = 913443 d = 53 

11 217 



log. of 8193217= 6.913455 371 

63 
106 



11.501 



PROBLEM 2. 

TO FIND THE NUMBER CORRESPONDING TO. ANY GIVEN 

LOGARITHM. 

52. By referring to the proportion of art. 51, and putting 
the value of x for the fourth term we have 

diff. of num. dif!. of logs. difT. of num. diff. of logs. 

100 : 000051 : : 42 : 000021 
• W« add 1 for the f^- rq'ected which is moi) 



48 PLANE TRIGONOMETRY. 

Instead of the 42 being given and the 000021 required as 
before, the 000021 is now given and the 42 required. 

The first term of the proportion is 100 or 1000, (fee, and the 

second term is in the column marked d, to find the third term 

multiply the extremes and divide by the second term 

.„ 000021 X 100 
42 = 

000051 
Hence the following 

RULE. 

To find the number corresponding to any given logarithm. 

Seek for the decimal part of the given logarithm, and we 
shall be readily guided to it, or else to the logarithm very 
near it, by means of the leading figures which are separated 
in the table from the others, to attract the eye. If we find a 
logarithm exactly agreeing with that given, then the number, 
which the table shows us to belong to the logarithm found 
will be the required number. If however, as is most likely 
we do not find the proposed logarithm exactly, then we are 
to take out the number corresponding to the next less loga- 
rithm; this number will of course fall short of that required, 
but the deficiency may be supplied as follows. Subtract the 
tabular logarithm from the given one, annex ciphers to the 
remainder at pleasure, and divide it by that number in the 
column D which is opposite to the tabular logarithm, and 
annex the quotient to the number already taken from the 
table. 

N. B. Should there be a quotient figure without annexing 
a cipher to the dividend, this quotient figure must be added 
to the last figure of the number taken from the table. Should 
it be necessary to annex two ciphers before obtaining a quo- 
tient figure, a cipher must be placed in the quotient and 
annexed with the figures that come after to the number taken 
from the table. 

The logarithm next greater than that given may be taken 
"^^om the tables, and the latter subtracted from the former in 



OPERATIONS BY LOGARITHMS. 49 

which case you would subtract the quotient obtained by 
dividing the difference as above instead of adding it. 

EXAMPLE. 

Find the number the log. of which is 5.868789 

The decimal part of the next less log. is that of 7392=.868762 

Their difference is 27 

Annex ciphers to this diff. and divide by the number in co- 
lumn D which is 69. 

59)2700(45 
^236 ^ 



340 
295 



Annex the quotient 45 to the number 7392 before found, 
and you have the number required corresponding to the given 
logarithm, namely 739245. This number contains 6 figures, 
one more than the characteristic of the given logarithm. In 
every case a sufficient number of ciphers must be annexed to 
obtain quotient figures enough, when appended, to make the 
whole of the number which thus results contain one more 
figure at least than is expressed by the characteristic of the 
given logarithm. If more quotient figures still be obtained, 
they will occupy the place of decimals. 

EXAMPLE 2. 

Find the number of which the log. is 2.913455 
Next less log. that of 8193 = 913443 

Number required is 819.322 53) 1200 (22 

106 

T40 
106 



EXAMPLES IN MULTIPLICATION AND D' 

LOGARITHMS. 

53. Required the product of 26784 and 



Bd FLANE TRIGOMOMETRT. 

log. of 26784* is 4.427875 

log. of 7.865 is . . 0.895699 

Their sum is 5.323574 
6.323574 is the log. of 210656, which last number is, there- 
fore, the product required. 

EXAMPLE II. 

Required the product of 3.586, 2.1046, .8372 and .0294 
log. of 3.586 . is . 0.554610 
of 2.1046 is 0.323170 

of .8372 . . . 1.922829 

of .0294 . . 2.468347 



Product 185761 . . . 1.268956 



EXAMPLE III« 



Divide 28.654 by 127.34 

log. of 28.654 . is . L457186 
of 127.34 . . . 2.194965 

difference 1.352220 

1.352220 is, therefore, the log. of the quotient which from 
the tables, observing the converse rule for pointing off 
decimals according to the characteristic, {4 art. 50) is 
.225019. 



* In looking for the log. of this number, look first for that of 2678, multiply 
the tab. di£ by 5^ the last figure of the given number, and cut off one figure iiom 
the product. 



OPERATIONS BY LOGARITHMS. U 

• EXAMPLE IV. 

Divide .06314 by 007241. 

log. of .06341 . is 2.800305 

of .007241 . . . 3.859799 



auotient 8.71979 . . 0.940506 

54. We shall now demonstrate rules for raising numbers 
to powers, and for extracting the roots of numbers, by n^eans 
of logarithms. 

Resume the equation, 



d='n 



raising both members to the w** power, we have, observing 
the rule of Algebra, which is to multiply the exponent by the 
degree of the power. 



a'** = n~ 



from this last equation, it appears that Vm is the power to 
which it is necessary to raise the base a in order to produce 
n* ; hence the following 



RULE. 



To raise a number to any power, by means of logarithms, 
multiple/ the logarithm of the given number by the exponent 
of the power, and the product will be the logarithm of the 
power. 

EXAMPLE I. 

Required the 4th power of .09163 

log. of .09163 is 2.962038 
Multiply by 4 

Product 6.848152 

5.848152 is the log. of .0000704939, wh r 

power required. 



53 t»LANE TRIGONOMETRY^ 

EXAMPLE II. 

Required the tenth power of .64. 

log of .64 . . . 1.806180 

10 



Power .0115292 2.061800 

In muhiplying the first decimal place by 10, the product is 

80, then 8 times 1 is 10, and 8 to carry is 2. 

55. To find a rule for extracting the root of a number by 
means of logarithms assume again the equation. 



a^ = w 



Take the m'* root of both members applying in the first 
member the rule to divide the exponent by the number 
expressing the degree of the root, and there results. 

_ tn 
I 

~ is here plainly the logarithm ot v w j hence the fol- 
lowing 

RULE, 

To extract the root of a number by means of logarithms, 
divide the logarithm of the given number j by the index of 
the root, and the quotient will be the logarithm of the root, 

EXAMPLE I. 

Required the 4th root of .434296. 

log. of .434296 -. , . 1.637786 

i of this logarithm is obtained by observing that the index 
which alone is negative, must be divided separately, as we 
should divide a minus term, followed by a plus term in 

Algebra ; the T can be rendered divisible by borrowing 3, and 
afterwards carrying +3 before the 6, rendering it 36 ; that 
^'s, the proposed logarithm is viewed imder the forna 

3.637786, 



LOGARITHMIC flINB8| &C. 63 

The quotient is r.909446,which is the logarithm of .811795, 
the fourth root required. 

EXAMPLE II. 

Required the 10th root of 2. 

log. of 2 . . . 0.301030 

Divide this by 10 . 0.030103 quotient, 

which is the log. of 1.07177, the root required. 

EXAMPLE III. 

Required the cube root of .00048 
log. of .00048 . 4681241 

i of it . . 2:893747 = log. of .078297, the root 

TABLE OF LOGARITHMIC, SINES, TANGENTS, &C. 

56. This is Table II. It contains the logarithm of the sine, 
tangent, cosine and cotangent, corresponding to every degree 
and minute in the quadrant.* 

These logarithms are those of the trigonometrical lines in 
a circle, the radius of which is 10000000000, or the tenth 
power of 10, the common logarithm of which is 10. As the 
sine is never greater than radius, its logarithm will always be 
less than 10, except for the arc 90^, the sine of which is equal 
to 10. 

PROBLEM. 

57. To find from the table the logarithm of the sine, tan- 
gent, cosine or cotangent of the number expressing any arc 

Case I. If the given number is composed of degrees ar 



• Without this table we should have been obliged to two o* 

tables which have been already described as follows. Fi *^^^ fo 
the natural sine, tangent, &c. of the g;iven arc or angle, u ^ 

this have entered Table I. and found its logarithm. 

9 



B4 PLANE TRIGONOMETRT. 

minutes, seek first for the number of degrees among those 
which are written at the top or bottom of the pages ; at the 
top if it is less than 45° ; at the bottom if it is greater. Run 
the eye down the first column which goes on increasing from 
top to bottom, if the number of degrees is found at the top of 
the page ; or up the last column which goes on increasing 
from the bottom upwards if the number of degrees is found 
at the bottom ; run the eye, I say, through one or the other of 
these columns in the direction in which it increases until you 
have found the number of minutes given ; upon the same 
horizontal line with the minutes thus found you will find 4he 
logarithm of the sine, cosine, tangent or cotangent which you 
seek. In order not to mistake the column, it is necessary to 
consult the title at the head of the column, if the number of 
degrees given is at the top of the page, but if it is at the bottom^ 
the inferior title must be consulted. 

EXAMPLE r. 

Required the logarithmic sine, tangent, cosine and cotan- 
gent of 190 55'. 

I find 19° at the top of page 37, 1 descend the first column 
at the left marked m which goes on increasing downwards till 
I find 65' ; upon the same horizontal line, and in the column 
entitled sine at top I find 9.532312, in the column entitled 
cosine 9.973215, in the column of tangents 9.559097, and in 
that of cotangents 10.440903 ; and these numbers are there- 
fore the numbers required. 

EXAMPLE II. 

Required tlie logarithmic sine and tangent of 70° 10'. 

I find 70° at the bottom of page 37, 1 ascend the last column 
marked m at bottom which goes on increasing upwards ; I 
find 10' in that column ; upon the same horizontal line I find 
in the column marked sine at bottom 9.973444, and in the 
column marked tangent at bottom 10.442879, which are the 
logarithms sought* 



TABLE OF 8iNE8, &C, 65 

Case II. If the given number is composed of degrees 
minutes and seconds, find the logarithm of the degrees and 
minutes as above, and then to know how much this should 
be increased for the given number of seconds, in case of the 
sine or tangent, or diminished in case of the cosine or cotan- 
gent, observe that the number in the cohimn marked d is the 
increase of the logarithm for I", and if multiplied by the given 
number of seconds the product will be the quantity to add to 
the logarithmic sine or tangent before found, or to subtract 
from the logarithmic cosine or cotangent. 

The number in the column d is calculated by subtracting 
one of two consecutive logarithms in the table, which differ 
by r from the other, and dividing the remainder by 60, the 
number of seconds in a minute ; the quotient is the difference 
of logarithms corresponding to a difference of 1" in the num- 
bers to which they belong. A decimal point must be under- 
stood two places from the right of each number in the column 
D. This calculation depends upon the principle mentioned 
at art. 51, that the differences of logarithms are proportional 
to the differences of their corresponding numbers. 

The logarithmic sines and cosines have each their column 
of differences annexed, but the tangents and cotangents have 
but one between them, the reason of which will appear from 
the following demonstration. 

By art. 37 we have 

tan X cot = R^ 

applying logarithms to this equation, since the log. of a pro- 
duct = the sum of the logs, of the factors, and the log. of a 
power = log. of the number raised to the power multiplied 
by the index of the power, we have 

log. tan + log. cot = 2 log. r = 20 
log. R being 10. Therefore having two arcs a and 6, sir 
log. tan + log. cot. in both is 20 we have 
log. tan a + log. cot a = log. tan b + loj tri 

posing 

log. tan a — log. tan b = log. cot a 
that is the difference of the logarithmic ta 
is equal to the difference of their logarx 



66 PLANE TRIG0N0M4ETRY. 



EXAMPLE I. 



Required the logarithmic sine of 40° 26' 28". 
I find the log. sine of 40° 26' to be 9.811952 ; the number 
in the next column d is 247, which I multiply by the given 
number of seconds 28 ; the product, inserting the decimal 
point which is understood to cut off the 47 in the number 
taken from the column o, is 69.16, which added to the loga- 
rithm before found 

9.811952 

69.16 



Gives 9.812021 rejecting the 16 

which goes beyond the usual number of decimal places. 

The logarithmic tangent of any given number of degrees, 
minutes and seconds, is found in a similar manner from the 
colunm entitled tangent. 

EXAMPLE II. 

Required the logarithmic cosine of 3° 40' 40". 
I find the cosine of 3° 40' to be 9.999110 ; the tabular dif- 
ference in the adjoining column is 13, which being multiplied 
by the seconds 40, the product is 5.20 ; subtracting* this 
result from 9.999110, the remainder is 9.999104, the loga- 
rithmic cosine sought. 

PROBLEM. 

To find the degrees^ minutes and seconds answering to 
any given logarithmic, sine, cosine, tangent or cotangent. 

The method is, of course, exactly the reverse of that just 
given. Look for the given logarithm in the proper column, 
which you will know from its title, either at the top or bot- 
tom, and if you find it exactly, the degrees will be found at 

• It will be recollected that as the arc increaees in the first quadrant, the cosine 



TABLE OF SINES, d&C 67 

the top of the page, and the minutes on the same horizontal 
line with your logarithm, in the first column at the left, if 
the title of the column is at top, but the degrees will be found 
at the bottom of the page, and the minutes in the column at 
the right, if the title of the column which contains your 
logarithm is at the bottom. 

If the given logarithm cannot be found, take the next less 
logarithm contained in the tables, subtract it from the given, 
annex two ciphers to the remainder, and divide by the num- 
ber in the coUuiin marked d ; the quotient is seconds, which 
add to the degrees and minutes belongixig to the logarithm 
found in the tables, if your given logarithm be that of a sine 
or tangent, but which subtract from the degrees and minutes, 
if a cosine or cotangent. 

EXAMPLE I. 

Required the number of degrees, minutes and seconds, of 
which the logarithmic sine is 9.88(J054. 

I find the next less logarithm in the column marked sine at 
bottom, to be 9.879963, which subtracted from the given 
logarithm, leaves 91 ; I annex two ciphers to this, and divide 
by the number 181, found in the column d adjoining; the 
quotient is 50, which is seconds. Taking the degrees from 
the bottom of the page, and the minutes from the column at 
the right, and in the same horizontal line with the logarithm 
9.879963, I have 49^ 20' 50' for the number required. 

EXAMPLE II. 

Required the number of degrees, minutes and seconds, of 
which the log. cotangent is 10.008688. 

I find the next less logarithm in the table, to be 10.008591 
that of 44° 26', which subtracted from the given logarithn 
leaves 97, to which annexing two ciphers, and dividir 
by the tabular difi*erence 421, the quotient is 23" ' ' tl 
required number is 44° 26' — 23" or 44° 25' 37" 

60. The secants and cosecants of arcs have 

10 



58 PLANE TRIGONOMETRY. 

serted in the table, because they may be easily computed from 
the cosines and sines. Thus : (Art. 33.) 



sec. 



r2 



COS. 

hence, - 

log. sec. = 20 — log. COS. 
for log. of the quotient = difference of logs. (art. 49 ;) and 
log. of the square of a number equal to twice the log. of the 
number, (Art. 64,) and log. of r = 10. 
To obtain the log. secant, therefore, we have this 

RULE. 

Subtract log. cosine from 20. 



R« 



Also, (Art. 35,) 

eosec. = 

sui 

hence, 

log. cosec. =20 — log. sine 

whence this 

RULE. 

To obtain log. cosec, subtract log, sine from 20. 

EXAMPLE. 

Bequired the log. secant of 48° 35' 27" 
log. cos. 48° 35' 27" = 9.820485 
log. sec. 48° 35' 27 ' = (20 — 9.820485) = 10.179515 

EXAMPLE II. 

Required log. cosec. 36° 27' 24". 

log. sin. 36° 27' 24'\ = 9.763493 

log. cosec. 35° 27' 24" = 20-9.763493 = 10.236507 

A method of finding with greater accuracy the sine 
^nd tangent of a very small arc, or the cosine and cotangent 
of one near 90°, is pointed out at Art. 1 23. 

To find the trigonometrical lines of arcs greater than 90°, 
observe the rule at Art. 17. 



RIGHT ANGLED TRIANGLES. 



09 



SOLUTION OF RIGHT ANGLED TRIANGLES, WITH THE AID 

OF LOGARITHMS. 

EXAMPLE. 

61. Referring to the example of Art. 39, where the hypothe- 
nuse 

1010x15 

^~ sin 35c 
employing 10^^ as r, instead of 1, because the tables which 
we are about to use, are constructed with that value of r, we 
have, by the rules for multiplication and division of loga- 
rithms 

log. of 10»o = 10.000000 
add log. 15 = 1.176091 

11.176091 
Subtract log. sin 35° = 9.758591 

Remainder 1.4 17500 = log. of 26.15 = the hyp. 

EXAMPLE II. 

jnniuuift 




200 feet. 



Referring to Art. 41 where the height of the tower c is 

c = tan 300 X 200 
which becomes, when the radius 1 in the first term of the pro- 
portion is changed into 10^® 

tan 30° X 200 



10 



10 



we have, by applying logarithms 

log. tan 30O = 9.761439 
add log. 200 =2.301030 

12.062469 
Subtract log. 10^ <> =10 



2.062469 = log. 118 



60 PLANE TRIGONOMETRY. 

62. It is customary, where the subtraction of logarithms 
corresponding to the division of numbers is to be performed, 
to change this operation into addition by means of what is 
called the arithmetical complement of each subtractive loga- 
rithm. The arithmetical complement of a logarithm is the 
remainder after takinsr the logarithm from 10 ; thus the arith- 
metical complement of the logarithm 2.322447, is 7.677553. 

10.000000 
2.322447 



7.677553 

It may.be formed most conveniently, instead of beginning 
at the right and subtracting each figure from 10 and carrying 
one each time throughout, by beginning at the left and sub- 
tracting each figure from 9 till you come to the last figure 
which subtract from 10. 

When we have a logarithm to subtract, we shall obtain the 
sume result by adding its arithmetical complement and after- 
wards subtracting 10. Which may be proved as follows : 

By the definition arith. comp. log. ^ = 10 — log. b. 

Now add this arith. comp. to some other log. as log. a, the 
result will be 

log a ~f- 10 — log b 

subtract 10 and there remains 

log a — log b 

The same result as would have been obtained by sub- 
tracting log. b from log. a. 

Hence to perform operations containing a number of multi- 
plications and divisions, by means of logarithms, we have 
the following 

RULE. 

Write the logarithms of all the multipliers, and the arith. 

complements of those of the divisors in a column. Add up 

'•e whole, and reject as many times 10 from the characteristic 

the sum as there have been arith. complements employed. 



RIGHT ANGLED TRIANGLES. 61 

Applying this rule to the last example but one (art. 61) we 

have 

log. 10^ = 10.000000 

log. 15 = 1.176091 

arilh. comp log. sin 35° = 0.241409 

Sum rejecting 1 from characteristic 1.41 75U0 = log. 26.15 
Applying it to the last example of Art. 61 we have 

log. tan. 30° = 9.761439 

log. 200 = 2.301030 

*ar. comp log. W = 0.000000 

sum rejec. 10 = 2.062469 = 115.4 
Applying logarithms to the example in art. 40, we have 
12^ + 16^ = 400 

log. of 400 = 2.602060 _ 

divide by 2, quot. = 1.301030 = log. V400 
From the tables 1.301030 is found to be the log. of 20 the 
value of the hypothenuse required. 

63. We shall finish the subject of right angled triangles 
by presenting a case of their practical application which is 
likely often to occur. 

77±J0U 




At the top of a mountain whose height was known by the 
barometer or otherwise to be 1000 feet above the level of the 
sea, a ship was observed through the tube of the instrument 
described at Art. 10, and the number of degrees between the 
tube and a plumb line from the centre of the circle was foui 
to be 77^ 30' ; required the distance of the ship. 

Aright angled triangle is here formed, in which are gi 
the perpendicular and angle at the vertex, bast 

required. . 

Referring to the rule r : tan. of one of th< 4ei 

side adjacf^nt : side opposite, we have in thi 
W^ : tan. 77^ 30' : : 1000 : dist. i 



♦ W^hcn arith. comp. of log. of r is used, it need not be wr 



62 



PLANE TBIGONOMETBY. 



To find the last term of a proportion, add the logarithms 
of the second and third terms which are the multipliers, and 
the arithmetical complement of the logarithm of the first 
term, which is the divisor, 
arith. comp. log. W = 0.000000 
log. 1000 = 3.000000 
log. tan. 77° 30' = 10.654245 

3.664246 = log. 4510.71 feet. 

Therefore the distance of the ship is 4510.71 feet, or a little 
over f of a mile. 

SOLUTION OF PLANE TBIANGLES IN GENEBAL. 

64. Let ABC be any tri- 
angle. From the vertex of 
one of the angles a, let fall 
upon the side opposite, the 
perpendicular ad; the given 
triangle will be divided into 
two right angled triangles 
ABO and ACD. 

In the first of these, by a proportion previously demonstrated 
(Art 38,) we have 

B : sin B : : A B : A D 
whence 

BXAD = sinBXAB 

Again, in the right angle triangle a c d, we have 

B : sin c : : A c : A i> 
whence 

BXAD = sinCXAC 

The first members of the two equations above, being the 
same, the second members are equal, hence 

sin B X A B = sin c X A c 

Turning this equation into a proportion, by xnaking the 
first product the extremes, and the second the means, we 




suiBcsm c:: Ac: A B 



OBLIQUE ANGLED TRIANGLES. 63 

That is, the sines of the angles of any plane triangle^ 
bear the same proportion to each other as the opposite sides. 

From the nature of the above demonstration the selection 
of the vertex from which to let fall the perpendicular being 
entirely unrestricted, it is plain that this rule applies to all 
the angles and sides alike. ' 

When therefore two of the three given parts of a triangle 
are opposite to each other, the part opposite to the third part 
may be found by the proportion which has just been estab- 
lished. 

We shall according to our custom suppose a practical pro- 
blem which introduces the case of solution in question. 

Let it be 

required to 
ascertain ac- 
curately the 

distance, 
from the 
town in the 
figure, across 

the impassable marsh, for the purpose of estimating the ex- 
pense of a causeway. 

Plant two staves with small flags, the one at the border of 
the marsh opposite the town where the causeway is to ter- 
minate, and the other at some convenient point from which 
the first and the town may both be seen. Measure the dis- 
tance between the staves, and let it be 1000 yards ; observe also 
the angle at the second staff by turning the tin tube of the 
instrument for taking angles first to the staff on the border of 
the marsh and then to the town; let this angle be 102°; 
observe also the angle at the town subtended by the line of 
the two staves, and let this last be 30°. A triangle will be 
formed in which are known an angle 3(P, the side opposite 
1000 yards, and another angle 102°, the side opposite to which 
is required. Then 

sin 30° : sin 1 02° : : 1000 : length of causeway required 




64 



PLANE TRfGONOMETRY. 



Arith. comp. log. sin 30^ 
(Art. 16) log. sin 102^ =log. sin 78° 

log. 1000 



0.301030 
9.990404 
3.000000 



3.291434=log.ofl956 
The length of the causeway must be 1956 yards. 

65. This proportion is also applicable to the case where two 
angles and the interjacent side of a triangle are given. Such is 
the case in the problem at Art. 10. 

One of the angles of that tri- 
angle being given equal to 47°, 
and the other equal to 105° 30', 
it will be easy to find the third 
angle, by recollecting that the 
sum of the three angles of a 
triangle is equal to two righC 
angles (Geom. B. 1, Prop. 25,) or 

180° ; therefore subtracting the 
sum of the two given, 47° + 105° 

30' = 152° 30' from 180°, the remainder 27° 30', is the third 
angle of the triangle, and is opposite the given side 500 yards ; 
hence 

sin 27° 30' : sin 47° : : 600 : dist. from It. ho. to ft. 
arith. comp. log. sin 27° 30 = 0.335594 
log. sin 47^ = 9.864127 
log. 500 = 2.698970 




2.898691= log. 791.9 
The distance from the light house to the fort is 791.9 yards. 
The importance, where any considerable degree of accu- 
racy is required, of the method of solution by calculation, 
instead of that by construction, will appear from the fact, that 
with tolerably accurate instruments, and some care in the con- 
struction, we made the required side, which we here find to 
'^e 791.9 yards, to be 800 yards upon the scale ; thus commit- 
g an error in the construction of about 8 yards. . 




OBtiaUB ANGLED TRIANOLES. 66 

66. We add another practical 

EXAMPLE. 

Involving the same case of solution combined with the 
solution of a right angled triangle. 

An observer upon a plain, desires to find the height of a 
neighboring hill above the level of the plain. 

Near the foot of the hill, 
let him take the angle of 
elevation to the top, and 
suppose it to be 66^ 64' ; 
then let him measure back a 
distance, say 100 yards, and anoyam 

again take the angle of elevation, which let be 33° 20'. Then in 
the triangle, of which 100 yards is the base, and 33° 20' is one 
of the angles at the base, we may have the angle at the ver- 
tex, and opposite to the given base, by observing that the ex- 
terior angle of a triangle being equal to the two interior 
and opposite (Geom. B. 1, Prop. 26, Cor. 6,) one of the 
interior is equal to the exterior minus, the other interior, and 
therefore the angle at the vertex here is equal to 66° 64' — 
33° 20' = 22° 34' ; then say as in the last example 

sin 22° 34' : sin 33° 20' : : 100 : side opp. to 33° 20' ; 
but in the right angled triangle of which 66° 64' is the angle 
at the base, and the height of the hill one of the perpendicular 
sides, we have the proportion 

10^® : sin 65° 64' : : hypoth. : height required, 
from which, multiplying means and dividing by the first term, 

sin 66° 64' x hypoth. 
height req. =; ^ — j-g^^ 

bat from the preceding proportion 

u *u 'A oooon/ sin 33° 20' X 100 
hypoth. or side op. 33° 20' = r— ^^^ Si, 

'^ ^ sm 22° 34' 

substituting this value in the last equation, we have 

^ . ^ . ^ sin 66° 64' x sin 33° 20' x 100 
height required = iQio x sin 22° 34' 

11 



66 PLAKft TfttMHOBCiraY. 

arith. comp. 10^® = 0.000000 

arith. comp. sin 22° 34' = 0.416942 

lo?. sin 55° 54' = 9.918062 

log. sin 330 20' = 9.739976 

log. 100 = 2. 



Slim rejecting twice 10 = 2.073979=log. 118.6* 
118.6 yards, or 366.8 feet, is the height of the hill. 



EXAMPLE III. 



67. Let there be a A 2ie^(it 

street in which the front 

of a triangular block 

is 216 feet, and another 

street makins: an angle 

of 22<> 37' with the first ; B 

under what angle must a third street be laid out from the 

extremity a of the first, so that the front of a complete 

row of buildings upon it shall be 117 feet in length ? 

8Qi;yTION. 




117 : sin 22° 37' : : 216 : the angle b 
. comp, of log. 117 = 7.931814 



arith. comp. 



»mp« 01 xog. 11/ = /.>fdiox4 
log. sin 22° 37' = 9.684968 
log. 216 = 2.334464 



sum rejecting 10 = 9.861236= log. sin iS^ 13' 65" 
=: B and 180° — b — c = 112° 9' ff' = angle a required. 

It must be observed that 9.861236 is also the log. sine 
of the supplement of 46° 13 56", because the sine of an arc 
is equal to the sine of its supplement. (Art. 16.) Hence 134° 
46' 6' is also a value of the angle b, and there are two 
solutions to the problem, which are both exhibited in the 
diagram. This case corresponds to Problem 11, B. 3, Geom. 

• To this ntolt tii« height of th« oyo or of tho instrumant should bo added. 



OBLtaVS AHOLCD TfttAjrOLXt. 



If the given angle were right or obtuse, there eonld be 
but one solution, and the required angle must be acute. 
. The same is the case if the given side opposite the givea 
angle is greater than the other given side, because in every 
plain triangle the greater angle is opposite the greater side. 

68. We shall next derive a formula for the solution of a 
triangle, when the three sides are given, and one or all of 
the angles required. 

Let A B c be any triangle ; JL 

from the vertex of one of the 
angles a, let fall a perpen- 
dicular A D upon the side op- 
posite. This perpendicular 
may fall either within or . 
without the triangle. First, -® 
suppose that it foils within ; then (Geom. B. 4, Prop. 12.) 




AC 



whence b d = 



« — AB«4-Bc2 — 2bcxbd 

A B« 4- B C« — A C« 



2bc 



Also in the right angled triangle a b d, we have 

R : cos B : A b : B D (Art. 38) 
whence multiplying the extremes and dividing by the third 
term 

R XB D 



cos B 



AB 



mibstitutingf in this last expression for b d itisi value obtained 
above, we have 

A b2 + B C« — A C« 

COSB = RX- o 

2 A B X B C 

aQ expression for the cosine of an angle in terms of the three 
sides of a triangle. 

Suppose now; that the perpendicular foils without the tri- 
angle. 



«9 



PLANS TRIGONOMETRY*. 



Then (Geom. B. 4, A 
Prop. 13) 

AC*= AB^+B c3+2 B DXBC 

hence 

AB»+BC^— AC^ 
" 2.BC J} 



— ^BD: 




Again, in the right angled triangle a b d 



R : cos a B D : : A B : B D 



hence 



cos A B D ==- 



R X B D 
A B 



But A B D is the supplement of the angle b of the triangle 
ABC, hence 

cos A B D = — cos B 

substituting — cos b for cos a b d above and changing the 
signs we have 



cos B 



rxbd 

A B 



substituting for — b d in the second member of this equation 
its value found above, we have as before 

A B^ + B c^ — A c9 

CO^^ = ^X 2ABXBC 

•or employing the small letters to represent the sides opposite 
the angles which are expressed by the large letters of the 
same name 



cos B 



€fi + c^ — b* 
2b c 



That is to say the cosine of either angle of a triangle is 
equal to the sum of the squares of the two sides which can- 
tain it, minus the square of the side opposite, divided by 
twice the rectangle of the containing sides, 

s 



Let us apply this formula to an 



OENERAL FORMULiE. 09 

EXAMPLE. 

Suppose the three sides of a triangular plat of ground arc 
to be 50, 60 and 70 yards, under what angle must the first 
two be laid out. 

. ^ , 2500 + 3600 — 4900 
cos required angle = r x __-^j_- _ = > 

if we make r= 1 ; hence i or 20000 is the nat. cosine of the 
angle required. This angle will be found from the table of 
natural sines and cosines, to be 78° 27' 47'.* 

This case might also be solved with the table of log. sines, 
&c. by subtracting the log. of the whole denominator from 
that of the whole numerator, and adding 10 the log. of r, or 
adding at once the arith. comp. of the denominator, the logs, 
of the numerator and r, rejecting 10 from the sum^, in either 
case the result would be log. cosine of the angle required. The 
solution is left as an exercise for the student. 

69. We proceed now to demonstrate some formulae which 
express relations between the different trigonometrical lines 
of the same arc, and between the trigonometrical lines of two 
different arcs. They are introduced here because necessary 
for the solution of the few cases of plane triangles which 
remain. We shall first derive formulse by means of which, 
when the sines and cosines of two arcs are known, the sine 
and cosine of their sum or difference may be found. Thus 
if the sine and cosine of 30° and also those of 20^ be given, 
those of 50° = 30° + 20° or of 10° =« 30° — 20° may be 
found. 



*Tbe seconds are found as follows : Take the differenco between the two < 
sines next greater and next less than yours, and also the difference beti 
yoon and the next greater, multiply the latter difference by 60 and divide the 
duct by the former difference; the quotient will be the seconds sought. The n 
appears from the following proportion, 
dit of cosines. dit of numbers. dif. of cosines. d* 

in the tab. : 60" : : yours and the tab. : f 

12 



ro 



PLAN£ TBIGONOMETRY. 



Let A M = a in the 
ram, be one of the 
given arcS} and b m = 
bf be the other. Then 
M p s= sin a, B I = sin 
bf since it is the perpen- 
dicular let fall from one 
extremity of the arc b 
upon the radius which 
passes through the other 
extremity, c p = cos 
a and c i the distance from the foot of the sine to the cen- 
tre = cos 6. AB = AM + M:B = a + &BE = siu (a + b) 
and E c = cos (a + 6). 

In the similar triangles c p m and c k i we have the pro- 
portion 




or 
or 



cm:mp::cx:ik 



R : sin a : : cos 6 : 1 K 



cm:cp::ci:ck 



R : cos a : : cos 6 : c k 



The triangle i b l is similar to c m p, because the two have 
their sides respectively perpendicular, * hence 



c M : c p : : B I : B L 



or 



1%: cos a : : sin 6 : B L 
also in the same triangles 



CM : MP : : BX : I L 



or 



R : sin a : : sin b : il 
Multiplying the means and dividing by the first term of each 
of these four proportions, we have from the first proportion 

sinacosfi 
IK = J (1) 



* TtspapeodieidK ndes ar» homologoui (Geom. B. 4| Propi 3]« SdioL) 



OENSmAL rOWLUVLM. Tl 

from the secondi 

cos a cos 6 

CK= ^ ... (2) 

firom the third, 

cos a sin 6 
BL = ... (3) 



from the fourth, 



smasm6 
IL= ... (4) 



Adding the eqations (1) and (3), we have from the addi- 
tion of the first members 

IK + BL which is = EL + BL = BE = sin AB = sin (a+b) j 
adding also the second members, the resulting equation ii 

. , , ,^ sin a cos 6 4- sin 6 cosa .^ 

sm(a + 6)= • • • (o) 

read thus : the sine of the sum of any two arcs is equal to 
the sine of the first into the cosine of the second plus th§ 
sine of the second into the cosine of the firsts divided Ay 
radius. 

If B = 1 the denominator of the second member ditap- 
pears. 

Subtracting equation (4) from (2) we have for the differ- 
ence of the first members 

CK — IL = CK — EK = C£==COS AB=s:C06 (a + 6) 

performing the subtraction also upon the second memdben 

there results 

cosacosft — sinasinft 
cos (a + 6) = . < . (6) 

read thus : the cosine of the sum of any two arcs is equal 
to the rectangle of their cosines minus the rectangle of their 
sines, divided by radius. 

In the last two formulee leta=60Oand 6 = 20^ then 
by the first 

. sin 60Q cos 20Q + sin 20^ cos 60 o 

'^ 1 or 10^^ as the case may be 



70 PLANE TRIGONOMETRY. 

by the second 

cos GQQ cos 20Q — sin 6 0^ sin 20 ^ 
cos 80° = 1 Of i(jio or whatever r may be 

We shall derive expressions for sin (a — b) and cos (a — 6) 

or the sine and cosine of the difference of two arcs in terms 

of the arcs themselves by making, in the formulae just derived 

for sin (a -{- b) and cos (a + 6) ; 6 = — 6, observing that cos 

( — b) = cos b and sin ( — 6) = — sin 6 (Art. 27). By this 

substitution there results 

sin a cos 6 — sin b cos a 
sin{a — b)= (7) 

and 

cos a cos b+ sin a sin 6 
cos (a — b)= • • • (8) 

70. These four formulae for the sine and cosine of the sum 

and difference of two arcs should be committed to memory, 

as they are constantly recurring in trigonometry and in the 

higher analysis. The four may be expressed in two by the use 

of the double sign, thus 

sin a cos b±sinb cos a 
sm (a i 6) = 

cos a cos 6 7 sin a sin b 
cos (a±b)= r 

71. From formula (5) sin (a + 6) = <fec., we derive one 
much used in the higher analysis for expressing twice an 
arc in terms of the arc itself, by simply making 6 = a the re- 
sult is 

. „ 2 sin a cos a ^ 

sm 2a = . . . (1) 

the two terms of the second member becoming the same. 

We also get an analogous expression for the cosine of twice 
an arc by making 6 = a in formula (6) of the last article 
cos (a + 6) = (fee. This expression is 

' cos^ a — sin* a ,^. 

cos 2 a = — ... (2) 

R 

Thus knowing the sine and cosine of 20°, these last two 
►jmulae would give us the sine and cosine of 40°. 



GENERAL FORMULA. 73 

These two formulas may be modified so as to express the 

sine and cosine of an arc in terms of half the arc under which 

last form they are much used. This is accomplished by 

making a=z ^ a, which is lef^itimate since a is supposed to 

have no particular value ; then 2 a becomes a and we have 

from (1) 

2 sin 1 a cos 1 a .ox 

sm a = ^ 2 — • . . (3) 

R ^ ^ 

and from (2) 

cos* ia — sin* fa ,*v 

cos a = = 2_ , , (4) 

R ^ ' 

72. By means of this last, and a very simple formula de- 
pending upon the well known property of the right angled 
triangle, that the square of the hypothenuse is equal to the 
sum of the squares of the other two sides, a formula ex- 
pressing the value of the sine of half an arc in terms of the 
arc itself may be obtained. 

The formula depending upon the property of the right 
angled triangle, will be found by referring to the last diagram, 
in which the triangle c p m is right angled at p, whence 
(Geom. B. 4, Prop. 11,) 

c M^ = c p^ + p M* 
or calling cm ^a 

cos« i a + sin« J a = r* . . (6) 

clearing equation (4) of the denominator and changing the 
order of the members it becomes 

cos' i a — sin«^a = RCosa . . (6) 
Subtracting (6) from (5) we have 

2 sin* J a = R* — r cos a 
dividing by 2 and taking the square root of both members of 
this last equation, we have the formula required. 

sin i a = V^ r^ — ^r cosa * ' (7) 

73. We resume the solution of triangles, having now a 
formula, by means of which we shall be able to derive an 
expression for one of the angles of a triangle in terms of the 



74 PLANE TRIGONOMETRY. 

three sides ; an expression which will be found much more 
convenient for the application of Ic^arithms than that con* 
tained in Art. 68. 
By Art. 68 

cos B = R pr ► . (1) 

2ac 
putting B in the place of a in the formula for sin ^ a (for* 
mula (7) of the last article) we have 

sin J B = VJr3 ^RCOSB 

substituting for cos b in this, its value in (1), we have 

' a a 2ac 

reducing the terms under the radical to a common denomi* 
nator, collecting them together and placing the common 
fiictor R* outside the parenthesis, there results 

sin i B = \/^i*^^H^EZ±ZJ 

4a c 

but (Alg. Art. 46) 

2 a c — a« — c3 =—{a—cy 

hence 

Aac i ac 

but the diflference of the squares of two quantities is equal to 
the product of their sum and difference, hence 

69 — (a — c)2 = (6 + a — c) (6 — a + c) 
substituting the second member of this in place of the first in 
the preceding equation, and separating the 4 of the denomi- 
nator into two factors 2 x 2, we have 

J ( {b+a — c) {b+c — g) 
sin Jb = R V < 2 2 

( be 

ut 

+ a — c b+a-^-c b+c — a b+c + a 
— g = —2 -cand— 2 = g « 

epresenting b + a + c the sum of the three sides of the tri- 



GENERAL FORMUUB. 7S 

angle by s^ the second members of the two la$t equalities 

become 

^s — c and ^ s — a 

substituting these for their equals in the preceding equation 
it becomes 



• 1 -v/ (i * — c)Cks — a) 

sm 4 B = R V — r^-^ ' 

be 

the formula sought. 

As the angles have each the same relations to the corres- 
ponding sides of a triangle, the same formula by a proper 
modification will furnish the values of the angles a and c. 

It may be expressed in ordinary language thus, fhesine of 
half either angle of a triangle is equal to radius into the 
square root of half the sum of the three sides minus one of 
the adjacent sides, into half the sum minus the other adjn^ 
cent side, divided by the rectangle of the adjacetU sides. 

To apply this to an 

EXAMPLE. 

Let there be three places at distances from each other 
respectively of 50, 60 and 70 miles. Required the angle 
under which two roads must depart from that which is 60 
and 70 distant from the other two, in the direction of these 
last. 60 and 70 will be the sides of a triangle adjacent the 
required angle, and 60 the side opposite then 

sin ithe angle =r V ^^ ^xTO 

1R0 

ij=t^= 9U,ij — 70=80 

and ^ « — 60 = 30 

log. of 20 = 1.301030 
log. of 30 =1.477121 
ar. comp. of log. of 60 = 8.22184P 
ar. comp. of log. of 70 = 8.1649( 

sum rejecting twice 10 =1.1649 
sum by % for V, quot. =>-^.6+774 



T6 PLANE TRiaONOMBTRT. 

*iLdd 10 to multiply by r sum = 9.677461=log. sin 
of ^ the required angle. 

From the tables we find J the angle to be 22^ 12' 28" 

The whole angle required will be double this or 

440 24' 56" 

74. Before treating of the only remaining case in the solu- 
tion of triangles, it will be convenient to demonstrate some 
additional general formulae which shall present certain import- 
ant relations of the trigonometrical lines of two different arcs ; 
which formulsB are of frequent use in the higher analysis, are 
employed in the subsequent parts of the work, and will be 
immediately of service in deriving a formula for the last case 
of plane trigonometry which we have to consider. 

Add together equations (5) and (7) of (Art. 69) which ex- 
press the values of sin (a + b) and sin (a — 6) and the 
resulting equation, cancelling the second terms of the second 
members which are similar with contrary signs, is 

sin (a + 6) + sin (a — b)=:- sinacosft ... (1) 

make a+b=p a.nd a — b=q 

add these last two equations ; there results 

2a = p + y whence a=:^{p + q) 
subtracting the same equations, the second from the first, 

26 =/> — q whence b= ^ (p — q) 
substituting in equation (1) the value of a-f 6, a — 6, a 
and b in terms of p and q, that equation becomes 

2 

sin p + sin y = — sin ^p +q) cos J(p — ?) • • • (2) 

Which may be translated into ordinary language thus : the 
sum of the sines of two arcs is equal to 2 divided by rad. 
into the sine of half the sum into the cosine of half the dif- 
ference of those arcs. 



* If the R be put under the radical, and made b*, the 20s log. of b^^ must be 
nduded in the column of numbers added up above ; this will cancel the twice 10 
rejected for two arithmitieal complements, and the whole operation will consist 
in adding up the column of four numbers as above^ and dividmg the sum. by 2. 



GENERAL FORMULAS. 77 

By subtracting the latter of the same equations (6) and (7) 
of (Art, 69) from the former, reducing similar terms, there 
results 

2 

sm (a + b) — sin (a — b) =- sin i cos a . . . (3) 

R 

making the same substitutions as above in equation (1) this 
last equation becomes 

2 

smp — smq = — sin^{p — q)cosi{p + q) ... (4) 

or, ike difference of the sines of two arcs is equal to 2 divi- 
ded by radius into the sine of half their difference into the 
cosine of half their sum. 

Divide equation (2) by equation (4) striking out the com- 

2 
mon factor ^ 

R 

sin p + sin q sin \{p + q) cos \ [p —q) 
sinjp — singr Q,os\{p •\-q)s\n\{p — q) ' * * w 
but 

!ilii (P±l) = ^^"g hiP + q) (Art. 32) and 
cos ^{p -\- q) R 

sm\{p—q) _ tang \{p - q) 

COs\{p — q) R 

or inverting this last 

cos \[p — q) _ R 

^m\{p—q) tani(p — gr) 

substituting in (5) the values of — %i;^~^l and ^. — a Y -} 
^ ^ ^ cos^(p+^) smi(p — 9) 

that equation becomes 

sin p + sin q tan \ {p + q) R 

sin p — sin gr r tan \[p — q) 

striking out r in the numerator and denominator the result- 
ing equation is 

sin p + s mq _ tan \ {p + q) 
sin p — sill q tan \{p — q) ' ' 

which may be expressed in a proportion thus : the sum oj 
sines of any two arcs is to the difference of f' ' 'net 
the tangent of half their sum, is to the tange 

difference, 

13 



78 PLANS TBIGOHOMETRT. 



'X 




76. Let ABC be any trian- 
gle ; then (Art. 64) 

a : 6 : : sin A : sin B 

% 

or by composition, (Oeom. ^- 

B. 2, Prop. 6,) ^i" 

a + b : a — fr : : sin a + sin b : sin a — nn b 
but by equation (6) Art. 74 

sin A + sin b : sin a — sin b : : tan ^ (a + b) :tan ^(a — b) 
hence, 

a + b : a — 6 : : tan ^ (a + b) ; tan ^ (a — b) 

That is to say, the sum of two of the sides of a plane trir 
mngle is to their difference as the tangent of half the sum of 
the opposite angles is to the tangent of half their difference. 

This proportion is employed when two sides and the in- 
cluded angle of a triangle are giren to find the other parts. 
Since the thr^ angles of every triangle are together equal to 
two right angles or 180^, subtracting the given included angle 
from 18(P, the remainder is the sum of the two angles oppo- 
site the given sides; then substituting for a and b in the above 
proportion the two given sides, three terms of it are known 
and the fourth may be found. After which, having half the 
sum and half the difference of the unknown angles, these 
angles themselves can be found by adding half the sum to 
half the difference for the greater, and subtracting half the 
difference from half the sum for the lesser ; when all the parts 
of the triangle will be known, except one side, which may be 
found by the proportion the sines of the angles are as the 
opposite sides. (Art. 64) , 



TmiAVaLBl IV OBlTBmAL. 



79 



EXAMPLE. 

An observer wishing to know the 
length of a small lake, measured two 
lines from the same point to the two 
extremities of the lake which he found 
to be respectively 153 and 137 yards ; 
he also observed with an instrument 
for taking angles the angle subtended 
from this point by the lake to be 4(P 
33' 12". 

SOLVTIOir. 

I. 

Tojind the two other angles. 

As sum of the given sides 290 

=arith.comp.of log, - ** 7.637602 

: diff. of the sides 16=log. - ^ 1.204120 

: : tan. of ^ sum op. angs. = 
j(180O— 40^33' 12") = 
690 43' 24" = log-» 

: tan. of ^ diffl of op. angs. 
log. of which is sum 
rejec. 10 == 




10.432446 



9.174168 =log. tan. 8^ 29' ST* 
Add and subt. with ^ sum of angs. 69^ 43' 24" 

sum = greater angle 78® 13' 1" 
diff. = lesser angle 61® 13' 47" 



• Thii noUtioa "szlog,-^i" ngptSmtokoH l^g, i$. 



80 



PLANE TRIGONOMETRY. 



II. 

To find the remaining side,* 

As sin 78° 13' 1" = arith.comp. log. - ^ 0.009249 

: opp. side 153t = log. - 1 2.184691 

: : sin 40° 33' 12" = log. - 1 9,813017 

: side opp. 40° 30' 12" = 

log. - 1 sum rejec. 10 = 2.006958 log of 

101.616 
The length of the lake is 101.616 yards. 

This case of the solution of a triangle combined with that 
exhibited at Art. 66, serves to determine the horizontal distance 
between two inaccessible objects. 

Let the distance between two towns which are in sight be 
required. 

Measure a line upon 
the ground (which is 
called the base line) of 2 
miles. Take the angles 
at each extremity formed 
by this base line and a 
line to each of the towns. 
Two triangles will be 
formed in each of which 
a side, viz. the base line, 
and two adjacent angles 
will be given. Let the 
angles in the triangle of 
which the upper town is 
the vertex be 159° and 
14° ; and those in that of 
which the lower town is 
the vertex be 25° and 
149°. Calculate the distance from one extremity of the base 




A more direct mode of finding tliisside when it is the only part required, is 
n at Art. 123. 

.53 is known to be the side opposite 78® 13' V because the greater angle of 
dQgle is always opposite the greater sido. (Geom. B. 1, Prop. 13.) 



SPHERICAL TRIGONOMETRY. 8t 

line, say the upper extremity, to each of the towns as in Art 
66. Then you will know two sides of a triangle the third 
side of which is the distance between the towns required. 

The included angle between these two sides is 169^ — 26° 
= 1340 

Having then two sides and the included angle, the re- 
mainder of the solution is the same as in the last case. 
We le^ve it as an exercise for the learner. 



PART IL 



SPHERICAL TRIGONOMETRY 



77, A spherical triangle is formed by three arcs of great 
circles. 

The planes of these arcs produced, form a trisolid angle, 
the vertex of which is at the centre of the sphere, (Greom. 
B. 9, Prop. 1.) 

The angle which two planes make with each other, is 
called a diedriil angle. This angle is equal to the angle 
formed by two lines, drawn one in each plane and perpen- 
dicular to the common intersection of the two planes at the 
same point. (Geom. B. 6, Def. 4.) 

The angles of a spherical triangle are the angles formed 
by the planes of the arcs, (Geom. B. 9, Def. 1.) These are 
the diedral angles of the trisolid angle, mentioned above. 

The sides of the spherical triangle are the arcs of the great 
circles, by which it is bounded. These arcs subtend the p' 
angles of the trisolid angle, and consequently measure 
The arcs are given in degrees, and since they contfl 
same number as the plane angles which thev subtenc 
plane angles may be employed in a deip ins 



82 



BPHSaiCAL TaiOONOMETaT. 



the ares, or sides of the spherical triangle ; and for a like 
reason, the diedral angles of the trisolid angle may be em- 
ployed instead of the angles of the triangle. 

If we suppose the trisolid angle, which has its vertex at 
the centre of a small sphere, to be produced so as to cut out 
a triangle upon a larger concentric* sphere, the sides of the 
triangle upon the larger sphere, containing respectively the 
same number of degrees as the plane angles of the trisolid 
angle, will contain the same number of degrees respectively 
as the sides of the spherical triangle cut out by the trisolid 
angle on the smaller sphere. So that as the number of de- 
grees in the angles and sides which are given or required, 
and not their absolute length, is taken into consideration in 
the solution of spherical triangles, the size of the sphere 
need not be regarded. 

78. Let ABC be a spherical 
triangle right angled at a. 
Let o be the centre of the 
sphere, and let the planes 
of the arcs which are the 
sides of the triangle be pro- 
duced so as to form the tri- 
solid angle whose vertex is 
At o. 

The plane angle cob will 
contain the same number 
of degrees as the side a of 
the spherical triangle; the. 
plane angle coa, the same 
number as the side b ; and 
AOB the same number as c ; so that these plane angles may be 
marked a, &, and c as in the figure. 

It has already been mentioned that the diedral angles of 
the trisolid angle correspond in the same manner to the 
angles of the spherical triangle ; and that these diedral an- 




• Uxnag the same teotrt. 



RIGHT ANGLED TRIANGLES. 83 

gles are measured by the angle of two lines, drawn one in 
each plain, perpendicular to the common intersection of the 
two planes at the same point. In order to draw these lines 
so as to be used most conveniently in the following demon- 
stration, take o M == the radius of the tables ; draw m p per« 
pendicular to o a, it will be perpendicular to the plane a o b ; 
(Geom. B. 6, Prop. 17.) since the two planes a o b and a o c 
are perpendicular to each other, a being by hypothesis a 
right angle ; from p draw p d perpendicular to o f, a 
line of the plane a o b ; join d m ; m d will be perpendicular 
to o B (Geom. B. 6, Prop. 6.) ; m d and d p being both perpen- 
dicular to o B at the same point d, the angle m d p is the died- 
ral angle of the planes a o b and Boc;orMDP = the angle 
B of the spherical triangle ; o m being equal to radius, m d it 
the sine of the plane angle a, and m p is the sine of the plane 
angle b ; in the triangle m d p, right angled at p, we hare the 
proportion (Art. 38.) 

R : sin D : : M D : M p 
substituting for d its equal b, for m d, its value sin a, and for 
M p, its value sin b, we have 

R : sin B : : sin a : sin 6 
That is, radius is to the sine of either of the oblique 
angles of a right angled spherical triangle as the sine of 
the hypothenuse is to the sine of the side opposite that angle. 

79. The solution of astronomical problems forms one of 
the most useful and agreeable applications of the theory of 
spherical trigonometry. To such inquiries the theory itself, 
no doubt, owes its origin, as well as many of the successive 
improvements which it has gradually received, so that a 
specimen of its use in the solution of astronomical problems 
may reasonably be looked for in a book on trigonometry. 

To illustrate therefore the above and subsequent formulsEt '^^ 
spherical trigonometry we shall accordingly introduce a 
great circles of the celestial sphere. They are so well km 
that to define them is perhaps superfluous. 

7%e equator is that great circle the plant* is 

pendicular to the axis of the earth. The t tf 

about which the earth performs its diam 



84 SPHBRICAL TRIGONOMETRT. 

produced to the celestial sphere becomes the axis of the 
heavens about which all the stars appear to revolve daily. 

The ecliptic is a great circle which makes an angle of 
about 23° 28' with the equator. It is the path which the sun 
appears to describe among the stars once a ySSir. 

The points in which the two great circles above defined 
intersect, are called equinoctial points. 

The one at which the sun crosses the equator in the spring 
about the 21st of March, is called the vernal equinox. 

The other which is where the sun crosses in the autumn, 
viz. about the 23d of September, is called the autumnal 
equinox. 

The meridians are great circles, the planes of which pass 

through the axis and the circumferences of which all intersect 

in the poles or points where the axis meets the surface of the 

celestial sphere. They are sometimes called hour circles. 

The sun appears to move about the earth once in 24 hours ; 

' 360° 
—--= 16° is the number of degrees through which the sun 

moves in an hour. 

The angle contained between the meridian of a place and 
tha,t meridian which passes through the sun at any given 
moment, is called the hour angle and converted into hours, 
15° to the hour, will show the time of day, if we reckon from 
noon instead of midnight as astronomers do. 

This time may be either a. m. or p. m. It is what is called 
apparent time, which varies a little from mean time, the time 
given by the clocks, in consequence of the slightly unequal 
motion of the sun in its annual revolution. 

The hour angle of a star is similar to that of the sun. 

The horizon of any place is a great circle whose plane 
touches the surface of the earth at that place, and extends to 
the celestial sphere. This is called the sensible horizon ; the 
real horizon is a plane parallel to this through the centre of 
the earth. When the sun or fixed stars are in question, the 
distances of which from the earth are so great that its radius 
as nothing comparatively, these two horizons may be 



CELESTIAL CIRCLES. 86 

regarded as coincident The zenith is Che pole of the horizon 
directly over head. The nadir is the opposite pole. 

The position of a heavenly body is fixed on the celestial 
sphere, like that of a place on the globe, by its latitude and 
lengiiude, only it must be observed that on the former these 
are measurexl from and upon the ecliptic instead of the 
equator. 

Similar measurements from and upon the celestial equator 
are called the declination and the right ascension ,the former 
corresponding to the latitude, axhd the latter lo the longitude. 

Longitude upon the earth is reckoned from some fixed 
meridian as that of Greenwich. 

Longitude upon the celestial sphere is reckoned from the 
vernal equinox which is called the first of Aries; right 
ascension also from the same point ; the former upon the 
ecliptic the latter upon the equator. 

The azimuth of a celestisd object is an arc of the horizon, 
comprehended between the meridian of the observer and the 
meridian which passes through the object. 

80. We are now prepared with materials for a practical 
application of the formulae of spherical trigonometry, and we 
commence with that already demonstrated. 

Let E in the annexed diagram 
be the equinoxial point, Ea a 
l)ortion of the equator, es a por- 
tion of the ecliptic, s the place of 
the sun, and sq, a portion of a 
meridian through the sun ; then 
sa will be the &s declination, bq, his right ascension, and es 
his longitude. 

Given the ©'s declination* equal to 20^, required hb longi- 
tude. 

In the right angled triangle eos right angled at a we know 
E = 23° 28' the opposite side sq, = 2(P required the hypo- 
thenuse eq. Hence the proportign 




♦ The declination of the sun may be found by takin* its meridian altitu^ 

14 wiUi 



8& 



SPHRRICAL TRIGONOMETRY. 



whence, 



R : sin 23° 28L : : sin es : sin 20^ 

R X sin 20^ 



sin ES 



sin 23° 28' 

arith. comp. log. sin 23° 28' = 0.399882 

lo<r, sin. of 20° = 9.534052 



log. of R = 10.000000 



sum rejecting 10 = 9.933934 = log. sin es 
Hence es = 59^ 11' 29' the longitude of the sun required. 
Let the student try the following modification of the pro- 
blem as an exercise. 
Given the 0's longitude equal to 90° to find his declination. 

81. By means of the proportion for right angled triangles, 
and of which an application has just been given, one may be 
derived for triangles in general. 

Let ABC be any spherical 
triangle ; let fall from a the 
arc AD perpendicular to the 
side Bc, the given triangle 
will be divided into two right 

and 



angled trianorles 



ABD 




whh the same instrument and in the same manner as was described at Art. ] 1. 
This observation should be made about noon repeatedly,and the greatest observed 
altitude will be the meridian altitude. A piece of colored glass will be required 
for the purpose. Let ^ be a 
place on the earth; pq its dis- 
tance from the equator will be 
the latitude ; this contains the 
same number ofdegrces as the 
arc ZQ' between the zenith and 
celestial equator. Let s be the 
Dlace of the sun, then sa will 

lis declination. Let no be 
horizon, then so is equal 

I meridian altitude, sz = 

oiiplement of his altitude, 

and is called the coaltitudc ; sot 

= Zd — sz or declination = 

latitude — coaltitude. 

N. B. The altitude of tlie uppermost point of the circumference of the sun 

aid be first taken, then of the lowermost point, and half their difference 

%i to the latter will give the altitude of the ©'s centre. 




TRIANGLES IN GENERAL. ST' 

ACD. In the right angled triai^gle abd we have tlie pro- 
portion (art. 78) 

R : sin B : : sin ab : sin ad 
and in the right angled triangle acd, the proportion 

R : sin c : : sin ac : sin ad 
Multiplying the extremes and means of each of these pro- 
portions we have the equations 

R X sin AD = sin b X sin ab 
and 

R X sin AD = sin c X sin ac 
The first members of these equations being the same the- 
second numbers are equal, hence 

sin b X sin ab = sin c x sin ac 
substituting for the sides ab and ac the small letters of the 
same name witli the angles opposite to them the last equa- 
tion may be written 

sin B sin c = sin c sin b 
or, 

sin B sin c 

sin h sin c 
or, 

sin B : sin 6 : : sin c : sin c* 
that is, the seines of the angles of a spherical triangle arc 
as the sines of the opposite sides, 

example. ^ 

Let z be the zenith, p the „. 

pole of the equator, and s the 
place of a star; zs will be 
the coaltitude of the star, 
zps its hour angle, ps its co- 
declination or polar distance, 
and szp the azimuth. Let 
the azimuth, coaltitude and 
hour angle be given to find 

the polar distance ; z, zs and p are given and ps reqoii 

«■ — — ■ ■ — — — ■ 

* The student will recollect that a proportion is an ec 
that ratio^ as commonly understood, is the quotient of tip 




% SPHERICAL TRIGONOMETRY. 

SOtUTION. 

sin p : sin z : : sin zs : sin ps 

Jjet p=32o 26' 6", z=49o 54' 38", and zs orp=U^ 13' 45" 

As sin p 320 26' 6'ar.comp. 0.270558 

: sin z 490 54' 38" . . . 9.883684 

: : sin p 440 13' 45" . . . 9.843563 

: sin z 84° 14' 40 ' . • . 9.997805 
polar dist. required. 

Since the sinQ of an arc is equal to the sine of its supple- 
ment (Art. 15.) the required side may be also the supplement 
of 84° 14' 40" or 950 45' 20". 

To illustrate this double solution by the diagram, let the 
student make or conceive to be made the following construc- 
tion. Draw an arc from s making with pz an angle equal 
to z, meeting pz in a point which we will call z'. sz' will 
then be equal to sz ; prolong pz and ps till they meet in the 
opposite pole which we will call p' ; a triangle will be formed 
z'p's, in which the angles z! and p', and the side sz' will be 
equ9.l to those given in the above example, but in which the 
jside p's is the supplement of ps.* 

"The polar distance of the fixed stars will be found to be 
1^1 ways the same, hence they describe circles about the poles 
in their apparent daily motion. 

82. We shall next demonstrate a formula which will ex» 
press one of the angles of a spherical triangle in terms of the 
three sides. 



* A rule for determining when there are two solutions, and when but one 10 
•Qcb CAseSi is given at Art 137. 



*» 



TRIANGLES IN GENEttAL 



89 



Let ABC be any spherical ^ 

triangle, o the centre of the 
sphere ; join oa, ob, oc ; a 
trisolid angle is formed ha- 
ving its vertex at o. The 
plane angle softhis trisolid 
angle may be called by the 
same letters as the sides of 
the spherical triangle for 
the reason given in Art. 77. 
By referring to the note of 
Art. 38, it will be seen, that 
we may choose at plesisure 
the length of a radius and 
the trigonometrical lines will 
have the same relations as 
those corresponding to the 
radius of the tables or any 
other radius. 

Let us take oa as radius and draw the perpeiidicuiats ad 
and AE at its extremity and in the planes aoc and aob ; pro- 
duce these perpendiculars till they meet the lines oc and ob 
in D and e. ad will be the tangent and od the secant of the 
side b of the spherical triangle, and Ae the tangent and oig 
the secant of the side c. ^ 

This being premised, let us take the value of de in terms 
of the other sides and one angle of each of the two plane 
triangles dae and doe, to both of which it belongs. This 

b^ + (? — a^ 




may be done by means of the formula cos a =2 r 



2 be 



(Art. 68.) from which taking the value of the square of th« 
side opposite the angle in the formula, we have 

2 b c cos A 



a^ = i8 -j- c* 



R 



in the triangle ead this formula becomes 



ED^ rsB Afi^ + ae^ — 



2 A D X ae cos A 



^m> 



16 



90 SPHERICAL TRIGONOMETRT. 

and in the triangle eod 

- 2 , 2 2 DO X Eo cos a 

ED9 = DO + EO^ 

R 

subtracting the former from the latter of these two equations 

and observing that in the right angled triangles oad and oae 

Do^ — AD^ = OA* and eo* — ae* = oa* 

there results 

^ „ „ 2DoXEocosa . 2adxaecosa 

= 2 o A* 

R ' R 

Taking the value of cos a from this equation, we have 

D O X E O cos a — R X O A^ 

cos A = 

A D X A E 

substituting for o a its value r for d o its value sec h = 

cos 6 
s 

(Art.33) forEo, sec c = -^ for a d its value tan b = ^ ^^" , 
^ '^ ' cose, . cosfr 

R Sin c 

(Art. 32) and for a e its value tan c = ^ the above ex- 

* cos c 

pression becomes 

— ,- X COS a — R^ R"^ COS a — R3 cos h cos c 

cos cos 

CPS A = 



Rsin& Rsinc r* sin b sin c 

cos b cos c 

or striking out r^ 

Rg cos a — R cos h cos c 
cos A = x-^ — ; 

sm'ft sm c 

But the angle a of the plane triangle d a e is the same as 
the angle a of the spherical triangle (Geom. B. 9, Prop. 6,) 
hence, translating the above fornmla, 

The cosine of either * angle of a spherical triangle is 
equal to radius square into the cosine of the side opposite^ 
minus radius into the rectangle of the cosines of the adja- 
cent sides, divided by the rectangle of the sines of the adja- 
cent sides* 

The above formula will servo to calculate one of the angles 

of a spherical triangle when the three sides are given if we 

* ■ ■ ■ ■ - 

We say either angle because in the above demonstration no particular 
was selected. 



TRIANGLES IN GENERAL. 91 

employ the table of natural sines and cosines ; but is unsuit- 
able for the application of logarithms, in consequence of the 
sign — in the numerator requiring a subtraction to be per- 
formed which operation is impracticable by means of loga- 
rithms. We shall therefore derive from this another formula, 
involving only multiplications, divisions, (fcc. of the trigo- 
nometrical lines contained in it, to which operations loga- 
rithms apply. 

It will be necessary first to establish another general rela- 
tion of trigonometrical lines which we shall require in the 
course of the demonstration. 

83. Resume the formulae for the cosine of the sum And 
diflFerence of two arcs (Art. 70.) 

cos a cos b — sin a sin b 

cos (a + 6) = 



cos (a — 6) = 



R 

COS a cos b -\-s\n a sin b 



R 

subtract the first from the second, substituting p for a + 6, 
and 5' for a — &> i (p + ?) f^r a, and ^ (p — q) for 6, as in 

(Art. 74) and we have 

2 

cos q — cos p = — sin ^ (p + q) sin ^ (p — ?) • • • (1) 

R 

That is, the difference of the cosines of any two arcs is 
equal to 2 divided by radius into the sine of half their sum 
into the sine of half their difference. 

84. Resume now the formula (Art. 72) 

sin i A = V ^ r2 _ ^ j^ cos A 
substitute in this the value of cos a in terms of the three sides 
of the spherical triangle given in the last article but one, 
(Art. 82) and there results 

. , -v / 1 o 1 R^ cos a — R cos b cos c /q\ 

smiA= V iR2_jR — Ti^T^^.; ^^ 

^ * sin sm c 

reducing the two terms under the radical to a common deno- 
minator they become 

R^ sin 6 sin c — r^ cos a + r^ cos b cos c 

2 sin b sin c 



92 SPHERICAL TRIGONOMETRY. 

collecting together the first and third terms of the numerator 

of this expression, the two may be written thus 

r2 (cos b cos c + sin 6 sin c) 

but 

cos 6 cos c + sin 6 sin o S&: it cos {b — e) 

(Art.69. . . . (8) ) hence the whole numerator of the fraction 

becomes 

r^ (cos (6 — c) — cos a) 

but by (1) above substituting b — c for ^f and a for p 

2 
cos (6 — c) — cosa= — sin J(rt 4-6— c)sin J(tf — b + c) 

R 

Multiplying the second member by b?; writing the denomi- 
nator 2 sin b sin c under it, striking out the common fiu^tors 
2 and r from the numerator and denominator, placing back 
the fraction thus reduced under the radical sign in equation 
(3), and esctracting the square root of r^ that equation becomes 

8miA = R Vsini(a + 6 - c)lR ^ (a + c- 6) _ ^^^ 

^ sm b sm c 

this may be a little modified by observing that 

i{a + b -c)=^{a+b+e)—c 
and 

i(a + c— 6) = i(a4-ft+c)~6 
representing a + b + chy s, these two expressions become 

i^s — c and J^s *^ b 
and equation (4) becomes 

1 *v/sin (is — b) sin (i s — c) 
sm * A = R V — ^* ; — L-. — \1 1/ 

sm b sin c 
read thus ; the sine of half either angle of a spherical tri^ 
angle is equal to radiiis into the square root of the sine of 
half the sum of the three sides of the triangle minus one of 
the adjacent sides^ into the sine of half the sum minus the 
other adjacent side, divided by the rectangle of the sines of 
the adja^cent sides. 

This formula, for the solution of a spherical triangle when 
^hree sides are given, is very convenient for calculation 
jarithms. Applied to each of the angles separately, it 
erve to determine them alL 



TRIANGLES IN GENERAL^ 93 

Take the following 

EXAMPLE. 

Let p be the pole of the ^ 

equator, z the zenith of the 
place of observation, and 
s the place of the sun. Then 
zp will be the colatitude, ps 
the codeclinalion, z s the 
coaltitude, and the angle zps 
the hour angle, or time of 
day, (apparent astronomical 
time). 

Given the latitude of a place of observation 30° 43' 37" 
the sun's declination 19^ 55' 42''j his altitude 26° 38' 33". 

Required the hour angle. 

colatitude = 90^ — the latitude = 69° 16' 23" 

codeclinntion = 90° — the declination = 70° 4' 18" 

coaltitude = 90°— the altitude = 63° 21' 27" 




The colatitude and codeclination are the 
sides adjacent the hour angle p. 

sum of the 3 sides = 192° 42' 8'' 

half sura = 96° 21' 4" 

subtract one of the adj. sides = 69° 16' 23'' 

remainder = 37° 4' 41" 
subtract from ^ s the other adjacent side = 70° 4' 18" 

remainder SE 26oi6'46" 

log. sin 37" 4' 41" = 9.780247 
log. sin 26«> 16' 46" = 9.6461"* 
ar. comp. of log. sin of one adj. side s= 0.066 
ar. comp. of log. sin of the other adj. side = 0.021 

sum rejecting twice 10 
16 



94 SPHERICAL TRIGONOMETRY. 

divide by 2 to take V, quot. = L759459* 
add 10 to multiply by r, log. prod. = 9.769459 
log. sin 35^ 4' 49" = ^ p. 

Hence p or th§ hour angle is 70° 9' 38". This converted 
into time, allowing 15° to the hour, is 4 hours 40' 38 ". 

In the same manner, and by the same formula, each of the 
other angles of the given spherical triangle might be found if 
required, taking care to use the two sides adjacent the angle 
about to be found each time. 

The result is not ambiguous, for sine of half the required 
angle must be acute since the whole angle cannot exceed 
180° (Geom. B. 9. Prop. 16. Schol.) 

86. It is proved in Geometry, (B. 9, Prop. 16,) that the 
three angles of a triangle being given the triangle is deter- 
mined. A formula for calculating either of the sides when 
the three angles fire given, may be easily derived from that 
of the last article, by means of the polar triangles. 

It is necessary first to premise that the polar triangles of 
the whole range of triangles will include all possible tri- 
angles ; for as each side of a triangle passes through all values 
from 180° to 0°, the opposite angle of the polar triangle will 
pass through all values from 0° to 180°. Wherefore what- 
ever can be proved of the polar triangles of all possible tri- 
angles, may be considered as proved for all triangles. 

Resume equation (4) of the preceding article. 

sni i A = R \ / sin ^ (g + 6 - c) sin \ (a +^^:=^) 

sin b sin c 

For the parts of the triangle in this formula, substitute 
their equivalents in the polar triangle. It will be remembered 
(Geom. B. 9, Prop. 8,) that each angle of a spherical triangle 
IS measured by a semicircumference or 180°, minus the side 



* To accomplish thir division add — 1 to the characteristic and -|- 1 to the 
decimal part of the logarithm, which additions balance each other and leare the 
value of the logarithm the same as before; the characteriptie thus becomei— S; 
\y !2I into — 2 = — 1 j.9 into 15 =; 7 j and lo on. 



TRIANGLES IN GENERAL. 95 

opposite in the polar triangle, and vice versa ; hence if a', 6', 
c\ represent the sides, and a', b', c', the angles of the polar 
triangle, we have 

A=180Q— a',a=180O— A', 6=180°— b' and c=18(y>— c' 
putting these values of the letters a, a, and c in their places 
in the formula above, it becomes 
sin \ (180O — a') 

^a W ^np80Ql^.(A+B'— c'))sin^(l8UQ---(r+c'— bQ) 
^ sin (180^ — B') sin (180° — c') 

But \ (180O— a')=(90O— i ^') and sin (90O— \ a')=cos \ a! 
(Art. 23); also ^ (180°— (a'+b'—c')=90o-4(a'+b'—c') and 
sin (90°— i (a'+b— c'))=cos \ (a'+b'— c'); also \ (180°— 
(a'+c'— b')=90°— J(a'+c'— b') and sin,(90°— J (a'+c'— b')) 
= cos \ (a'+c'— b') ; also sin 180°— b'= sin b' (Art 16) and 
sin(180°— c')=sinc. 

Making these substitutions, the formula becomes 

cos i a' = R \/ ^^" ^ (^' + ^' - ^^) ^ .^" * (^' + ^' - »') 

sin b* sm c* 

a formula for the cosine of half a side in terms of the three 
angles of a triangle. We may leave out the accents over the 
letters, which wc have employed only to distinguish the polar 
from the triangle to which it corresponds, and which are 
superfluous in a general formula. 

This formula will undergo a similar modification to that 
made in formula (4) of the last article, 

i(A + B — C) = J(^ + B + C) — C 

and 

i(A + C — B)=i(A + B+c)— B 

Represent a + b + c by s, and the second members of 
these two equations may be written 

j^ s — c and ^ s — b 

substituting them in the place of the first members in the 
formula, it becomes 

, ^ / cos (is — b) cos (As — c) 

cos i a = R V — — . — <—,--^ r 

2 sm B sin c 

or, the cosine of half either side of any spherical tria 



96 SPHERICAL TRIGONOMETRY. 

■ is equal to radius into the square root of the cosine of half the 
sum of the three angles minus one adjacent angle into the 
cosine of half the sum minus the other adjacent angle 
divided by the rectangle of the sines of the adjacent angles, 

86, We shall next derive two sets of proportions applica- 
ble to the solution of a spherical triangle, the first set when 
two sides and the included angle are given, and the second 
when two angles and the included side. For this purpose 
it will be necessary to establish some additional general rela- 
tions of trigonometrical lines. Resume from (Art. 70) the 
formuise for the cosine of the sum, and for the cosine of the 
difference of two arcs. 

cos a cos b — sin a sin b 
cos (a -h b) = ^ 

and 

cos a cos 6 + sin a sin b 

cos (a — 6)= 

Add these two equations together, cancelling sin a sin 6 
in the second member, and substituting p for a+6, and q for 
a — 6, \ (p+ j) for a, and \ {p — q) for 6, as in Art. 74 ; there 

results 

2 

Cosp+co8q=— cos ^{p+q) cos ^{p—q) ... (1) 

R 

By this last equation divide equation (2) of Art. 74. 

2 

sin p + sinq — sin ^ {p+q) cos ^ {p—q) 



R 



2 

cos p + cosq — cos \ {p +q) cos J (p— 5) 

R 

Striking out the common factors— and cos J {p — q) in the 
nomerator and the denominator of the second member, and 

the lost equation becomes 

sin p + sin g _ tan ^ (p-fg) 

cosp + cosq R • • • W 



GENERAL FORMULAE. 97 

Again, divide equation (4) of Art. 74, by equation (1) of 
this article, 

sin p — sin y - sin \{p—q) cosJ(p+5f) 



2 
cos p 4- cos g - cos ^ {p+q) cos \ {p — q) 

Aw 

striking out from the second members the common factorf 

2 

— and cos J (p-fqr), and observing that 

R 

s\n1^(p—q) tani(p—q)^ , . ^ 

CQsA(p— g) ~ R ^^ '^^^ equation becomes 

sin p — smq _ tan j (p—q) 

cosp + cosq R • • • W 

A^ain, referring to Art. 71, we find a formula (3) for tha 
sine of an arc, in terms of half the arc, viz. 

2 sin jl^ a cos i a 
sm a = ' ^ — 

R 

substituting ip+q) for a in this formula, it becomes 

2 

Sin(p+5r)=-sin J(p+5r)c0S^(;)+gr) . . (4) 

Xv 

By this last equation divide equation (2) Art. 74, 

sin ;? + sin ^ —sin ^ (p-^q) cos ^ (/?— y) 

= 2 
sin (p+q) —sin J (p+ur) cos ^ (p-^q) 

R 

2 
striking out the common factors — and sin ^ (p+y) in tha 

R 

second member, this equation becomes 

sin p +sin q cos^ {p — q) 

sin (p+W'~cosl{p+q) 
Again, divide equation (4) of Art. 74, by equation 
this article, we have 

sin p — sin y — sin ^ ip-^q) ' 

=2 

sin ip+q) — sin^ {p+q) 



(5) 



98 SPHERICAL TRIGONOMETRY. 

Striking out the common factors, as in the last case, this be- 
comes 

sin p — smq_ sin j {p—q ) . ^ .gj 

Making ^=0 in equation (1) of Art. 83, and observing that 
cos 0=R, we have 

2sm^^p 

R — cos p =: • 

^ R 

But (Art. 71) 

2 sin i « cos i p 
sin p = ^-^ ^-^ 

R 

Dividing the latter by the former, and striking out the 
common factors — and sin ^ p, we have 



R 

sin p cos ^ j9 



But (Art. 34) 



hence 



R — cosp sin^p 

cos^p cot J^ p 

sin J p R 



(7-) 



sinp cot ^ p 

R — COSp R 

87. Resume now the formula for the cosine of an angle in 
terms of the three sides of a spherical triangle, (Art 82.) 

r2 cos a — R cos b cos c 

cos A= : ; : 

sm sm c 
die same formula applied to the angle c is 

r2 cos c — R cos a cos b 

cos C = : — ; 

sm a sm b 
islearing both these equations of fractions, they become 

cos A sin 6 sin c = r^ cos a — r cos b cos c 
and 

cos c sin a sin b =k^ cos c — r cos a cos b 

"^iliminate cos c from these last two equatiojjs by the 
3d of addition and subtraction. The co-ciSicients of cos c 
ie rendered the same by multiplying the first equation 



TRIANGLES IN GENERAL. 99 

by R and the second by cos b. Performing this multiplica- 
tion, the two equations become 

R cos A sin 6 sin c = r^ eos a — r^ cos b cos c 
and 
cos b cos c sin a sin b = r' cos b cos c — r cos a cos* b 
Add these, cancelling the common term r* cos b cos c and 
there results 
R cos Asin 6sin c+cos 6cosc sin asin6=R^cosa — Rcosacos* ( 

substituting for cos^ b in the last term its value r* — sin* & 
(Art. 72) this result becomes 

R cos A sin b sin c+cos b cos csin asin 6=r3cos a — r' cosa+ 

R sin* b cos a 
cancelling 4- r' cos a and — r^ cos a in the second member, 
and dividing the whole equation by sin 6, we have 

R cos A sin c + cos b cos c sin a = r sin i cos a^ 
Transposing the second term of the first member, this last 
equation becomes 

R cos A sin c = R sin b cos a — cos b cos c sin a 
Had we set out with the value of cos b and combined it 
with that of cos c in the same manner that we have that of 
cos A, it is plain that we should have obtained a similar result 
which we may write without the trouble of repeating the 
operation, by changing a into b and a into 3, and vice rersa^ 
We thus have 

R cos B sin c = R sin a cos 6 — cos a cos c sin b 
Adding this and the preceding equation together we haver 
R sin c (cos A+cos b)=(r — cos c) (sin a cos 6+sin b cos a) 
but (Art. 69) 

sin a cos 6 + sin 6 cos a = r sin (a + 6) 

Substituting this second member for the first in the pre/ieding 
expression, and dividing the whole by r, we have 

sin c (cos A + cos b)=(r-— cos c) sin(a+6) . . . (1) 

But (Art. 81) 

sin c sin a 

sin c sin a 

which cleared of fractions becomes 

sin c sin A = sin c sin a . 



100 SPHERICAL TRIGONOMETRY. 

Also 

sin c sin b 

sin c sin b 
the denominator^ of which being made to disappear it be- 
comes 

sin c sin B = sin c sin 6 . . . (3) 
adding (2) and (3) we have 

sin c (sin a + sin b) = sin c (sin a + sin 6) . . . (4) 
subtracting (3) from (2) there results 

sin c (sin a — sin b) = sin c (sin a — sin 6) . . . (5) 
dividing (4) by (1) we have 

sin A + sin b sin c sin a -f sin 6 ^ /p\ 

cos A + cos B R — cos c siu {a -{- o) 
dividing (5) by (I) we liave 

sin A — sin b sin c sin a — sin 6 /y'x 

cos A + cos B li— cos c siu (a -f b) 
But it appears from equation (2) of the preceding article 

that 

sin A + sin r tan j^ (a + b ) 

cos A + cos B R 

from formula (3) of the same article that 

sin A — sin b tan ^ (a — b) 

cos A + cos B R 

from formula (7) of the same article that 

sin c cot ^ c 

R cos C R 

from formula (5) of the same article that 

sin a -f sin 6 cos ^ (a — b) 

sin (a + b) c'os'| (a^^b) 
and from formula (6) of the same article that 

sin a — sin b sin J (a — 6) 

sin {a -p 6) sm I (a -^ b) 
substituting these values in equations (6) and (7) of the pre- 
sent article those equations become 

tan i (A + B) = cot i c —i-^^^ 
■* ^ ^ ^ cos i (a + 6) 



TRIANGLES IN GENERAL. 101 

and 

tan i (a — b) = cot A c-.-^-, Ti.x 
^ ^ ' -* sin ^ (a + 6) 

These may each be written in the form of a proportion,* thus 

cos ^ (a + 6) : cos \{a — 6) : : cot J^ c : tan ^ (a + b) . • • (8) 

and 

sin ^ (a + 6) : sin ^ (a — 6) : : cot ^ c : tan ^ (a — b) . . . (9) 

That is, the cosine of half the sum of two sides of a sphe- 
rical triangle is to the cosine of half their differ ence^ as the 
cotangent of half the included angle is to the tangent of 
half the sum of the other two angles. 

The second may be repeated in a similar manner, change 
ing cosine into sine and tangent of the half sum into tangent 
of the half diflFerence of the other two angles. 

When two sides and the included angle of a triangle are 
given, the first three terms of both these proportions will be 
known, and the fourth in both, or rather its logarithm, may 
be found by adding the logarithm of the second and third, 
and the arithmetical complement of the logarithm of the first. 
We shall then know half the sum \ (a + b) and half the dit 
ference ^ (a — e) of the two unknown angles. By adding 
and subtracting these results the angles themselves will be 
obtained ; thus 

4 (a + b) + i (a — b) = A 

and 

i (a + b) — i (a — b) = B 

From the proportions (8) and (9), by means of the property 
of polar triangles, another set may be derived, which shall be 
applicable to the solution of a spherical triangle, when two 
angles and the interjacent side are given. Thus in the pro- 
portions (8) and (9) change each of the sides and anglos into 
180^ minus the side opposite in the polar triangle. 

cos i (a + 6) becomes cos \ (180^ — a + 180^ — b) 
cos ^(1800 — (a + b)) = ^ cos i (a + b) 



*m» 



• The last term of a proportioii being eqaal fo U^ pi 
|eirms4ivi(l^ bjr |he firau 



SPHERICAL TRIGONOMETRY. 



since the cosine of the supplement of an arc is equal to minus 
the cosine of the arc. (Art. 25). 
Cos i (a — 6) becomes cos ^ (ISPP — a — (180^— b)) = 

cos J ( A + B) = cos ^ (a — B) 

since ( — a + b) = — (a — b) and since the cosme of a 
negative arc is positive (Art. 27.) 

Cot i c becomes cot ^ (180°— c)= cot (90°—^ c)= tan ^ c 
since the cotangent of the complement is the tangent. 
In a similar manner 

tan ^ (a + b) becomes — tan ^ (a + 6) 

The two extremes of the proportion by these substitutions 
become negative ; their product will still be equal to that of 
the means, if they be both made positive ; proportion (8) thus 
becomes 

cos J^(a -{- b) : cos ^ (a — b) : : tan ^ c : tan ^ (a + 6) 
By similar changes proportion (9) becomes 

sin i (a + b) : sin ^ (a — b) : : tan ^ c : tan ^ (a — b) 

The last one may be translated into ordinary language 
thus: 

The sine of half the sum of two angles of a spherical 
trig^ngle is to the sine of half their difference as the tangent 
of half the interjacent side is to the tangent of half the sum 
of the other two sides. 

The other may be repeated in a similar manner. 

These two proportions, and the two from which they were 
derived, are known by the name of Napier's analogies,* 
having been first given by Lord Napier, who is celebrated 
for many useful inventions of a similar character, but chiefly 
fo? that of logarithms; 

We shall now apply the first set to an 



EXAMPLE. 



The latitudes and longitudes of two celestial objects being 
'ven to find their distance apart. 



A.nalogy is a term synonymous withproportioQ. The first term bears tho 
s analogy or proportion to the seooiid that the third does to the fourth. 




TRIANGLES IN. GENEBAL 108 

Let p be the pole of the ecliptic, 
s and 8 the places of the two ce- 
lestial objects, then ps and ps' 
will be their coaltitudes, and the 
angle p will be the difference of 
their lonoritudes, since p will be 
measured by that arc at a quad- 
rant's distance on the ecliptic 
(Geom. B. 9, Prop. 6.) Let the 
latitudes of the two celestial ob- 
jects be 51° 30' and 20° ; and let 
their difference of lonjritude be 31° 34' 26''. Their colati- 
tudes will be 38^ 30' and 70o. Then we shall know in the 
above triangle the two sides opposite s and s' which we will 
call s and s'j and the included angle p. The greater side j^ 
= 70° s = 38° 30' and p = 3P 34' 26". Applying Napiert 
analogies with the use of logarithms, we have by the first 
ar.comp.log.cos.^(5'+5)or54oi5 =0.233401 
log.cos.|(y— 5)orl5o45'=9.983381 
log.cot.4porl5o47'13"=10.548G35 

sum rejecting 10 = 10.765417=log.tan.J(s'4-») 
i (s' + s) = 80° 15' 41" 
Had ^ {s + s) been greater than 90^, its cosine must have 
been negative, and the first term of the proportion being nega* 
tive the fourth must have been negative also, and i (s' + a) 
would have been the supplement of the angle found in the 
tables, since the tangent of the supplement is equal to minus 
the tangent of an arc. (A.rt. 36.) 

Applying the second proportion we have 

ar.comp.log.sin.^(5'+5)or 54^15 =0.090672 

log.sin.^(V— 5)orl5045'=9.433675 

log.cot.^porl5°47'13"=10.548635 

sum rejecting 10 = 1 0.072982=log.tan.i(s'— s) 
^ (s' — s) = 49° 47' 30" 
Add 

i (s' +s') = 80° 15' 41" 

sum = 8' = 130O 3' 11" 



104 SPHERICAL TRIGONOMETRY. 

Subtract the upper from the lower. 

Difference = s = 30° 28' 11" 
Since we know now all the parts of the triangle except the 
side p opposite the angle p, that might be found by the pro* 
portion, (Art. 64,) the sines of the angles are as the sines of 
the opposite sides. But to avoid an ambiguity in the result 
similar to that of Art. 81, and the trouble of determining which 
of the two results corresponds to the other parts of the trian* 
gle now fixed, it is better to employ one of the second set of 
Napier's analogies, inverting it, thus 
ar.comp.log.cos.i(s'— s)or49°47'30"=0.19C067 
Iog.cos.^(s4-s)or80°15'41"=9.228281 
log.tan.^(5'+5)or54oi5'* = 10.142730 

sum rejecting 10 = 9.561068 = log.tan.^p 
^ p=t20^ and /?=40° the distance, between the celestial ob- 
jects, required. 

A method of determining the unknown side at once, if that 
be the only part required when two sides and the included 
angle are given, is exhibited at Art. 126. 

If two angles and the interjacent side of a spherical triangle 
Mrere given, the method of proceeding would be to employ the 
second set of Napier's analogies in the same manner that we 
have the first in the example above. 

The student may take the following example as an exercise : 

Given the declination of a planet, the angle which its path 
makes with the meridian through it, and the difference •of 
wderialt time between its passage across a given meridian at 
the lime the declination was observed and some subsequent 
epoch, to find its declination at the latter epoch and the dis- 
tance that it has moved. 

♦ This log. need not be found from the tables, but may be obtained by sub-« 

Iractiflg the ar. comp. of sin i (5' -f- 5) above from ar. coYtip. cos J (s'-j- 5) and 

A sifi 

«iddinfl: lO ; for since tan. = , log. tan. = 10 + log. sin — log. cos. Call- 

° ' cos • o o 

the ar. comp. log. cos. and s ar.comp.log. sin, this becomes 10 + (10 — s) 

— C) r= 10 + C — 5. Q. E. D. 

derial time is measured by the daily motion of the fixed stars, aad does 
ry like solar time with the annual motioo of the sua. 



RIGHT ANQLBD TRIANGLES. lOS 

Liet p in the last diagram be the pole of the equator ; s the 
first position of the planet, and s' the last ; ps or s will be the 
given codecUnation ; p the given difference of time, and s the 
angle which the planet's path makes with the meridian ; ps 
or s will be the codecUnation, and ss' oip will be the distance 
moved. 

Let p = 2 hours or 30°, s = 70°, and the interjacent side 
s' = 50° to find s and p. 

88. The formulae for the solution of spherical triangles in 
general which have now been demonstrated, apply of course 
to right angled triangles ; but if it be recollected that the tri- 
gonometrical lines of the right angle or 90° are either r, 0, 
or 00, it will be evident that these formulse may, when thus 
applied, be much simplified. 

The student can easily make the substitutions necessary to 
change the foregoing formulae into such as apply exclusively 
to right angled triangles, for himself. We shall not occupy 
space with them here, but be content with observing, that 
after they have been made, all the formulae which result will 
be found capable of being expressed in two short rules, or 
these indeed may be united into a single one. 

Amongst all the convenient and useful inventions of 
mathematicians, none is more ingenious and beautiful than 
this, the author of which is the celebrated Lord Napier, whose 
name we already have had occasion repeatedly to mention 
in connection with the most happy discoveries for facilitating 
mathematical operations. The rules are known as 

Napier's rules for the circular parts. 

The circular parts of a right angled spherical triangle are 
~The two sides including the right angle, called 

1. The base. 

2. The perpendicular* 



And 



3. The complement of the hypothenuse, 

4. Tiie complement of the angle at the 
6. The complement of the angle at tb 




106 SPHERICAL TRIOONOMETRT. 

The right angle being entirely left out of consideration in 
the solution of triangles of this kind, the angle at the base is 
that included between the base and the hypothenuse ; and the 
angle at the vertex is that included between the hypothenuse 
and perpendicular. 

The circular parts are then the parts of the triangle itself 
except the right angle, only that the complements of the hypo- 
thenuse and oblique angles are the circular parts instead of 
these themselves. 

The annexed diagram shows of which parts the comple- 
ments are used. 

There being five of these circu- 
lar parts, itisevident that any three 
of these which you choose to select 
will either be contiguous, or else 
two will be adjoining and one will 
be separated from them by a part 
on each side. 

In the first case, the part intermediate between the other 
two is called the middle part, and they are called its adjacent 
parts. In the second case, the part which is separated from 
the other two is called the middle part, and they its opposite 
parts. By means of this arrangement, all the relations of a 
right angled triangle may be expressed in the two following 
rules of Napier : 

1. Radius multiplied by the sine of the middle part is 

equal to the rectangle of the tangents of the adjacent 
parts. 

2. Radius multiplied by the sine of the middle part is 

equal to the rectangle of the cosines of the opposite 

parts. 

Or both rules may be given thus : radius into the siiie of the 

middle part = the rectangle of the tangents of the adjacent 

parts = the rectangle of the cosines of the opposite parts, 

"'he memory will be aided by observing that the words 

iuts and adjacent in the second clause of the above 

)Oth contain the letter a ; and that the words cosines 

)pposite in the last clause both contain the letter o. 

' the right angle of a right angled spherical triangle is 



RIGHT ANGLED TRIANGLES. 107 

always known, any other two parts being given, the rest may 
be found. 

The method of proceeding, is as follows : Take the two 
given parts and one of the required parts, or if but one of 
the unknown parts be required, lake that, you will thus have 
under consideration three parts of the triangle. One of these 
three will be middle, and the other two either adjacent or op- 
posite ; apply the rule of Napier, and you will have an equa- 
tion resulting which will contain the two given parts and the 
required part ; make the required part the unknown quantity 
in the equation and resolve it, you will thus obtain the value 
of the required part in terms of the two which were givea. 
By applying logarithms to this value, you will have it in de- 
grees, minutes, and seconds. 

EXAMPLE I. 

89. Given the sun's right ascension and declination to 
find his longitude. 

Let the parts of the right an- 
gled spherical triangle eqs repre- 
sent the same circles of the ce- 
lestial sphere, as at Art. 80, cal- 
ling the sides opposite the an- 
gles by the small letters of the 

same name with those at the angles, 5 will be the right as- 
cension, e will be the declination, and q the longitude, s and 
e are given and q is required. 

Of these three parts 5, e, and y, s and e are contiguous, and 

q is separated from them by a part on each side ; therefore q 

is the middle part and 5 and e are opposite parts. Applying 

Napier's rule, remembering that the complement of the hyr 

thenuse q is to be employed, we have 

R X sin of comp. q = cos s cos e 

or, 

R COS q = cos s cos e 

whence, 

cos s cos e 
cos5f= 




108 SPHERICAL TRIGONOMETRY. 

Suppose the sun's right ascension, on the I7th May, to be 53^ 

38' and his declination at the same time 19^ 15' 57 ", then 

cos 530 38' X cos 190 15 57" 
cos q = ^^iQ 

ar. comp. log. W^ = 0.000000 

log. cos. 530 38' = 9.773018 

log. COS. 190 15' 57 " = 9.974971 

sum rejecting 10 = 9.747989 = log. cos. 55^ 67' 43" 
Hence q or the longitude required is 65° 57' 43". 

EXAMPLE II. 

90. The same being given as in the last example, required 
the obliquity of the ecliptic. 

The required part is the angle e in the figure. Of e, s and 
e, since the three are contiguous leaving out the right angle, 
a is the middle part, hence applying the rule of Napier, 

R sin ^ = tan e cot e 

We put cot E instead of tan e because, according to the 
directions before given, the complements of the oblique angles 
are to be employed. Taking the value of cot e from the 
above equation, we have 

Rsin^ 

cot e =- • 

tane 

Applying logarithms to the second member, we have 

ar. comp. log. tan e 19° 15' 57" = 0.456521 

log. sin s 530 38' = 9.905925 

log. R == 10.000000 

sum rejecting 10 == 10.362446 = cotE 
E= 23° 27' 50"* 

91. Had the angle s been required, it might be found by ob- 
serving that of the three parts, s, e and 5,the latter two of which 



9 obliquity of the ecliptic is continually, though very slowly, changing. It 
/a given with the miauieat attainable accuracy in the Nautical Aimanaci 



RIGHT XNGLED TBIXN0LE8; tOS 

are giren, e is the middle part, and s and s adjacent. Henee^ 
using the complement of the oblique angle is,- 

R sin e = tan ^ cot s 
wh^ce 

R sin e 

cot 8=-T— — 

tan s 

Appl]ring logarithms as before, the reitle df s #iH he foandf. 
We leave this as an exercise for the leameif. 

In the solution of the above triangtei it y^ be obeenred 
that we have found each of the unknown pairts in terms Df 
the two given, and have not employed! oM 6f those first cpH* 
culated to obtain another. This is agreeable to the prii^tlpt^ 
laid down at Art. 41, of plane trigonometry, and the reason 
is the same. Such a method of proceeding is always prac- 
ticable in the solution of right angled spherical triang^. 

In the examples which we have taken above, we bant 
supposed the base and peipendieatar of m tij^t angled 
spherical triangle given. Any other two par£^ l&eing given^ 
each of the unknown parts may be dklculated' 6y the aid of 
Napier's rules, in a manner entir^ shRiUiar to what has been 
just exhibited. 



19 



r 

HO SPHERICAL TRIGONOMETRY. 

EXAMPLE III. 

^2. Given the sun's declination to find the time of his 
rising and setting at any place whose latitude is known. 

Let fiE s a represent the 
mefictian ef the place, z being 
tbe zenith, and ho the horizon, 
iHd let s* s" be the apparent 
path of the sun on the proposed 
day, catting the horizon in s. 
Then the arc ez will be the 
latitude of the plade, and con- 
seqiiehdy eh, or its eqUal ao, 
will be the colatitude, sind this 

nfe^eusures the angli^ OAa ; afso rs wilf be the sun's declina- 
tion/ and AR, expressed in time, will express the time of sun- 
rise from 6 o'clock, for nA^s is the six o%l6ck hour circle. 

Hence, in the r^ht-angled triangle sar, we have given rs 
dnS the opposite an^te a to find ar, the time from 6 o'clock. 

EXAMPLE u. 

Required the time of sunrise at latitude B2P 13* N., when 
the sun's declination is 23P 28'. 
By Napier's riJe, 

Rad. sin ar = cot a. tan Rs=:tan. lat tan. dee. 
tan 23^28' - - 9.637611 

tan 52 13 . - - 10.110579 




sin 34 3 21^" - - 9.748190 

4* 



Aiiintime2*16'13"25' 
6 



3 43 46 35= time of rising. 



degrees are converted into hours by multiplying by 4 an^ diriding b^ 
bich is equivalent to multiplying by 15. 



AIGHT ANGLED TRIANGLES. Ill 

SCHOLIVm. 

It should be b^e renf^arked that the time thus determine^ 
is apparent time, which is that which would be -shown by 9, 
clock so adjusted as to pass over 24 Jiours during one appar 
rent revolution of the sun, or from its leaving the meridiai^ 
to its Teturn to it again, the index pointing to 10, ^vlien (he 
sun is on the meridian. But it is impossible that any dock 
can be so adjusted, because the interval between the sa<y 
cessive return of the sun to the meridiian is continually iicary- 
ing, on account of the unequal motioa oflhe sun in jls.whit, 
and of the obliquity of the ecliptic^, each lof these varyiog 
intervals is called a true solar day^ and it is the mean of these 
during the year which is measured by the 24 hours of a well 
regulated dock, this period of time beting a jmdn t^lar day j 
hence, at certain periods of the year, ,the sua will acrive 1^ 
the meridian before the clock .poiots to 12, and at lOthe^ 
periods the clock will precede Ihe sun ; the small interval 
between the arrival of the index of the clock to 12 and of the 
sun to the meridian is called the equation of time, and it if 
given in page ii. of the Nautical Almanac for every day in 
the year ; this correction, therefore, must always be applied 
to the apparent time determined by :trigonometrical /^tjLcul^ 
tion to obtain the true time, or that shown by a wdl regulate^ 
clock or chronometer. 

Another circumstance too mu^ be taken into account, 
in order to determine the apparent time with rigorous Accu- 
racy, viz. the change in the declination of the sun from suit- 
rise to noon. In the Nautical Almanac, the declination of fl^e 
snn is given for every day at noon, and if this be used in the 
computation we shall assume that the declination has j^ot 
varied from sunrise to noon, which is not the case ; tience 
it will be necessary to compute the declination for the time 
of sunriee, as determined ahove^ and then to resolve the pr 
blem with .this corrected declination. The cprrection is { 
tained by taking from the Nautical Almanac the variatioo 
declination in 24 hours, and then finding by >n 

variattion ixc the time required. 



m 



9»UE^4GAh TaiOONOICETRY^ 



9. Required the time of sunrise at latitude 67^ 2' 54", wfaea 
{the sun's 4^1in9,tion }s 23P 28' ? 

3* IV 49" 



VROBLEM III. 

93. Given the latitude of Che pM^ee, jand the declination of 
a heavenly body, to determine its altitude and azimuth when 
OQ tbt six o'clock hour circle. 

|>t EDSFO i)fi the meri4ia;Q 9f 
tlm pllice, z tkb z^th, HO the 
horijEon, s the place of the ob- 
ject OQ the sjz (^^oek Jiowjcicr 
ole Fsp, which of course passes 
thiough the east and west 
jpoiilits of the horizon^ ai;i4 zsb 
fibe yertical ^ir^le passing 
tim^Vi^ the suiPL. Then, m 
fy^ right-an^ed triangle sBiii^ the gi^ren quantities are as, the 
dedination, and the arc op, or aogie sab, the latitude of the 
place, to find the iijititude BSf, ai^d the azimuth bo from the 
fliorth poinjt of the hiori|x>xi ; 0r to find the complement ab of 

is a^ninth, thiit is, the sua's ^tearing from the east. 




p:|LMIPLES. 

1. What was the altitude and azimuth of Arcturus, when 
«p<m the six o'clock hour circle of Greenwich, lat. 61^ 28' 
40" N., on the 1st of April, 1822 ; its declination on that da; 
beingau^6'60''N.$ 

By Napier's rule we have 

Rad. sin bs = stn a sin as 

Rad. cos A 



Rad. cos a = tan ab cot as .*.* cot bo =• 



cot AS 



* This sign .*. tignifies ifurffan. 



RIGHT ANGLED TRX ANGLES, 119 



For the altitude. 
dnA 61o%'40" 9.893410 
sin AS 20 6 50 9.636416 



sin Bs 16 36 27 9.429826 



For the azimuth, 
cos A - - 9.794361 
cot AS r - 10.436265 



cot BO 77^9' 4" 9.358106 






Hence the alUtude is 150 36' 27', and the azimuth 77^ 9' 4" Jf, 
94. There remains one case in the solution of oblique 
angled spherical trianp^les, which we have deferred to this 
place, because we wished to employ in it the rules fi>r tbs 
solution of ri^t angled triangles. 

This is where two sides and the angle opposite* to one of 
them, or two angles and the side opposite to one of them are 
given ; or as it is sometimes expressed,where two of the giveii 
parts are a side and its opposite apgle, In such a case we 
may proceed as follows. 

By means of the proportion, the sines of the angles arfi of 
the sines of the opposite sidfis (Art, 81), the unknown par( 
opposite one of the given parts may be found. Four parts 
of the triangle will then be known, and two will remain 
unknown ; these two will be a side and its opposite apgle, to 
find which, from the vertex of the unknown angle let fall an 
arc perpendicular upon the unknown side opposite, and the 
given triangle will be divided into two partial triangles, which 
will be right angled and in each of which two parts will be 
known. Applying Napier's rules to the solution of these, the 
partial angles which compose the unknown angle may be 
found, and their sum will be the value of the unknown angle ; 
then the unknown side opposite may be found by the pro- 
portion, the sines of the angles are as the opposite sides ; or 
this last side may be found by calculating the two parts of 
which it is composed from the right angled triangles, and 
adding them together, which is the better method since it 
avoids ambiguity. If the perpendicular arc, drawn from tim 
vertex of the unknown angle to the unknown side falls with- 



• The term opponte is used in a more eztct ■enee here ihar llii 

•f which we have juit heon speaking. 



114 SPHERICAL TRIGONOMETRY. 

out the triangle, of course the dijBference, instead of the sum 
of the angles and sides found in the right angled triangles, is 
to be taken. 

Thus in the annexed triangle 
let the side and angle opposite 
given be c and c, and the other 
giveii angle b. Then first 

sin c : «in .«".:: sin s: sin i 
by means of which proportion b 
may be calculated and will be 
legitimately ambiguous. Then 
there witl be left unkiiowii a and a. From a let fall a per- 
pendicular A B lipon a ^hieh we ha^e «ot drawn, lest it 
should cenfuse the diagram, but which the student can im- 
agine ; then in the right angled triangle b a d, we know two 
parts B and c, and also in the right angled triangle c a d we 
know two parts c and 6, the latter having been found by the 
proportion above. To ca^xilate the partial angles at a, call- 
ing th«,t ^ .the ^rst right .aiHgM ti^iaogle above mentioned <», 
Mxni that in 4he second w', we have by Napier's r^^les 

B cos X = cot B ^Ot Of 

whence 






„ . R cos c 

COl CM = 

«0t6 




and in the 


same manner 

^ , fi cos * 

^©t w 

oet<; 




ibeR 


«» -^f- iw' X 




«nd 








sine : sin« :r, ski a *: 


Anf€L 


whence 


Bin c sin a 

sina= -. 





«m c 
Thus all the pa* ts *of the triangle «re Aetewnioed, 



RIGHT ANGLED TRIANGLES. If 6 

For an application of this case of soltttiotv find the angle 
s and the side za in the example of Art. 81. 

R cos js , R cos p 

^^^ *" = -^STT ^^= -cot z 

z = 84oi4'40"' » + »' = » 

p = 44^13'45" 
P = 32^26' 6" 

z = 49054'38'' sin ssinp- 

^= — i 

sm p 

We have now demonstrated formulae for the sohttion of 
every possible case of plane andsphencal triangles, including 
the more simple formula which apply exclusively to the 
right-angled triangles. We have had occasion for this pur- 
pose to derive many formulse expressing the general relations 
of the trigonon^etrical lines. A treatise has thus been formed 
complete in itself, and containing nothing that could weU be 
omitted. For greater convenience in committing to memory^ 
Tce give in the next Part^ a recapitulation of the results.* 



IIG 



RECAPITULATION. 



PART III. 



]tECAPITULAT](»r. 



i 



General formate expressing the relations of trigonometrical 
lines* 



(») 
(2) 

(8) 

(6) 
(6) 
*7) 

(8) 
(9) 

m 
(11) 

(12) 
(13) 



• . . . cos a = sin (90<> — a) 
. • . .' sin a = cos (90^ — a) 
. . . . sin a = sin (180<> — a) 
. .' • . sin a = — sin ( — a) 
.cos (180° — o) = — cos a 
. • . . cos a = cos ( — a) . 



sinf = Vr« — co8« 



cos = VR* — sin' 

Rsin 
tan = 



cos 



tan co8^ 

smf = — -. — t 



cot = 



R 

R cos 



sm 
(an X tot = R^ 



sec 



»3 



cos 



(14) cosec = 



sm 



(Art. 83) 
(Art. 83) 
(Art. 16) 
(Art 87) 
(Art.8g) 
(Art. ST) 
(Art. 72) 

(Art. 72) 
(Art. 32) 



(Art. 34) 
(Art. 37) 

(Art. 33) 
(Art. 36) 



* Some few of the general formulsc which have been derived in the cotix'fle o^ 
work are here omitted i* all tbat are found in thia recapitulation, may witlif 
itage be committed to memory', 
rhia it derived from the precediag^ 



RIGHT ANGLED PLANE TRIaNOLES. 117 

/ic\ xn 2 sill a COS a ,* «.,v 

(16) • . . • siii 2 a = . . . (Art. 71) 

/\a\ 2 sin i a COS A a ,^ ^,^ 

(16) • , . . Sin a = ^ ^— . . (Art. 71) 

/lr^ co^^a — sin^^a ,._,vtx 

(17) . . . cos a = ^ . . (Art. 71) 

(18) . . • sin J a = v^RS — jacosa . . (Art 72) 

/iQ\ • / I IN sin a cos 6 + sin 6 cos a , ^ ^^^ 

(19) • sm (a + b)=: = (Art. 70) 

/on\ / _j_ rx COS a cos 6 Z sin a sin 6 , ^ 

(20) . cos (a + i) = ' R (Art. 70) 

2 

(21) sin p ± sin y = — sin i (p + g) cos J(p + g) (Art 74) 

2 

<22) cos /> + cos jr = -- cos J (p + q) cos i{p — q) (Art. 86) 

2 
<23) cosp — cos jr = - sin J (p + 5) sin ^(p — g) (Art. 83) 

R 

(24) . . , sinp + sin , _ UB J (p4^,) 

^ ' . Sin « — sin n titn X (n — m\ v-^*** ^y 



sin p — sin 5 tan i (p "" ?) 



or, 



.sum of the sines : cfe^. o/*/Ae ^ne^ : : ian ^ sum : tan ^ di/" 
ference. 



IL 



Formulae and rules for the solution of triangles* 
1. Right angled plane triangles. 
A being the right angle and a the hypothenuse. 

CASE I. 

The hypotheniise being either given or required, and one 
of the angles. 

(1) . . R : sin B : : a : 6 or R : sin c : : a : c . . . (Art. 38.) 
i. e. Had : the sine of either acute angle : : the hypothenuse : 
the side opposite the angle, 

CASE II. 

The hypothenuse being neither given nor requ' 
<pne of the acute angles being either given or reqi 

19 



1 18 RECAPITULATION. 

(2) . . R : tan b : : c : 6 or r : tan c : 6 ; c . . . (Art. 41) 
i. e. Rad : tan of either acute angle : : the side adjacent thai 
angle : the side opposite* 

In the second of proportions (I) and (2) sin and tan of c 
may be changed into cos, and cot of b. 

CASE III. 

When two of the sides are given and the third required. 

a = Vprp^ or 6 = v a" — c" or c = v ^8 _ ja 
2. Oblique angled plane triangles. 

CASE I. 

Two of the three given parts being a side and its opposite 

angle 

sin A : a : : sin B : 6 : : sin c : c . . . (Art. 64) 

which may be expressed thus 

sin A a , sin B b . sin a a 
-. — = -- ana — — = - and - . — = - 
sin B b sni c c sm c c 

or the sines of the angles are as the opposite sides. 

This rule when used alone if an angle be required gives 

two solutions since sin of an arc = sin of its supplement. If 

otlier parts of the triangle besides the four in the proportion 

are already determined, they serve to indicate whieh of the 

two is to be chosen. 

CASE II. 

Two angles and the interjacent side being given to find 
the other parts. 

Subtract the sum of the ttoo given angles fronjL ISO^, ih» 
remainder is the third angle. Then say as sine of this 
angle : the given side opposite : : the sine of either of the 
other angles : the side opposite. 

CASE III. 

!*he three sides being given to find the angles 

«n ^ 4 ^ > y/W^b) (i^-c) ' ' (Art. 73.) 

b e 



ftlOHT ANGLBD SPHERICAL TRIANGLES. llO 

or the sine of ^ either angle of a trianffle = rad. into the 
square root of \ the sum of the three sides — one of the sides 
adjacent to the angle^ into \ the sum — the other adjacent 
sidey divided by the rectangle of the adjacent sides. 

This applied to each augle separately will determine all 
three. 

CASE iy» 

Two sides and the included angle being given to find the 
other parts. 

Let A be the given angle and b and c the given sides, then 
b + c :b — c : : tan i (b + c) : tan ^ (b — c) . . . (Art. 7f .) 
or the sum of the two given sides : their difference : : the tan 
of\ the sum, of the two unknown angles : the tan. of\ their 
difference. 

B + C=180O— A 

Three terms of the above proportion being known, the 
fourth, tan J (b — c), may be found, then 

\ (B + c)+i(B — C) = B 
and 

i(B + c)-~i(B — c)=c 

All the unknown parts will thus be determined, except the 
side opposite the given angle, which may be found by the 
proportion. 

The sines of the angles are as the opposite sides. 

We add one other formula for the cosine of an angle in 
terms of the three sides for its miscellaneous use, rather than 
to be employed in the solution of triangles. 



cos A== R 



2 be 



SPHERICAL TRIGONOMETRY. 

1. Right angled spherical triangles. 

Napier^s Rules. 

R X sin of the middle part s=: the rectangle c 
of the adjacent parts = the rectangle of xht 
opposite parts. (Art. 88.) 



120 RECAPITULATION^ 

Observe that the complements of the hypothennse and ob^ 
lique angles are used. 

The middle part is either between two others which are 
adjacent to it, or else separated from two which are adjoin-' 
ing each other. 

N. B. The two given parts must always be employed with 
one of the required parts in applying these rules ; by this 
means an equation will be formed from which the value of 
the required part contained in it must be derived^ 

2. Oblique angled spherical triangles, 

CASE I. 

The three sides being given to find tFie angles^ 

.;r. i . » / sin (i s — b) sin iiJs — cT 
8miA=RV sin6sinc ' ' i^^^'> 

Or the sine of J either angle of a spherical triangle = ra- 
dius hUo the square root of the sine of ^ the sum of the 
three sides minus one of the sides adjacent the req^uired 
ans^lei into the sine of \ the sum minus the other adjacent 
sidcj divided by the rectangle of the sines of the adjacent 

sides. 

This rule applied to each of the angles will serve to deter- 
mine them all successively. 

CASE II. 

Where the three angles are given 



^ sm B sm c ^ ' 

or the cos. of J either side of a spherical triangle = radius 
into the square root of J^ the sum of the three angles m,inus 
one of the angles adjacent the required side into ^ the sum 
minus the other adjacent angle, divided by the rectangle 
"ines of the adjacent angles. 

B rule the three sides may be separately found. 

CASE III. 

iides and the included angle being given, employ 



RlatlT ANGLED SI^HERK'AL miANGLEa. IZI 

Napier^s Analogies. 

cos i (a + 6) : cosi(a — ft) :: cotjc : tani(A + B) 
6in i (a + ft) : sin J (a — ft) : : cot J c : tan J (a — b) 

i(A + B)+i(A-B) = A 

4(a+b) — 4(a— b) = b 
The first proportion is read thus : 

Cos of J the sum of the two given sides : cos of ^ their 
difference : : cot, of^ the given included angle : tan. of ^ 
the sum of the unknown angles. The other is read in a simi^ 
lar manner. 

To find the remaining side 

sin A : sin a : : sin c : sin c 
or to avoid ambiguity 

cos J (a — b) : cos j^ (a + b) : : tan i (a + ft) : tan \ c 
observe that log. tan. J (a + ft)= 10 + ar. comp. log. co8.| 
(a+ft) — ar. comp. log. sin \{a + b) to save one reference to 
the tables. . . . (Art. 87.) 

CASE IV* 

When two angles and the included side are given. 

Napier^ s Analogies. 

cos ^ (a+b) : cos ^ (a — b) : : tan J c : tan J^ («+&) 
sin J (a+b) : sin J (a— b) : : tan ^ c : tan ^ («— ft) 

h («+6) + i («— ft) = « 

i (a+ft) — i (a-ft) = ft 
sin a : sin A : : sin c sin c 
or better 
cos J (a— ft) : cos J^ (a+ft) : : tan J (a+b) : cot J c. (Art. 87.) 

CASE V. 

Two of the three given parts being a side and its opp 
angle. 

Find the part opposite the other given part bv the p 
sition, the sines of the angles are as the Tie i 

site sides, i 



122 RECAPITULATION. 

Then there will remain unknown in the triangle, an angle 
and the opposite side. 

Let fall from the vertex of the unknown angle, an arc 
perpendicular upon the unknown side, the given triangle 
will thus be divided into two right-angled spherical triangles, 
in each of which the hypothenuse and one angle are known, 
apply Napier*s rules to find each of the partial angles 
which compose the required angle of the given triangle^ 
and the partial sides which compose the required side. 
This case is ambiguous, since there will be two values corres- 
ponding to the fourth term of the first proportion, " the sines 
of the angles are as ihe opposite sides." Both of these solu- 
tions answer to the conditions of the problem. 



We add the fundamental formula of spherical trigonome- 
try, that of the cosine of an angle, in terms of the three sides, 
which applied to each angle furnishes three equations, and 
the six parts of the triangle, from which e([uations any two 
parts may be eliminated, and the result will contain four parts, 
any three of which being given the fourth may be found. 

R^cos a — R cos ft cos c 

cos A = : — r — : • 

sm ft sin c 
or, ihe cosine of a spherical triangle is equal to radius square 
into the cosine of the side opposite minus radius into the rec- 
tangle of the cosines of the adjacent sides, divided hy the rec- 
tangle of the sines of the adjacent sides* 

Note. — In assuming hypothetical cases, care must be had 
lot to suppose such as are impossible. The following are 
he governing principles to be observed* 

In plane triangles, 1. One side must be less than the sum 

of the other two, (Geom., B. 1, Prop. 7). 2. The greater side 

of a triangle is opposite to the greater angle, (Geom., B. 1, 

"^ •^. 13). 3. The sum of the angles must be exactly two 

singles. 

pherical triangles, the first two principles also apply, 
3., B. 9, Props. 1 dnd 14). 4. The sum of the threa 



GEMKRAL FORMULiB. 183 

ati^les must not be less than two, nor greater than six right 
angles (Geom., B. 9, Prop. 16). 5. The sum of the three 
sides must be less than a circumference (B. 9, Prop. 3). 
6. Each side must be less than asemicircumference (B. 9, Def. 
1). 7. Each angle must be less than two right angles, (B. 9, 
Prop. 16, SchoL). 



Part IY. is a short treatise containing the application of 
plane and spherical trigonometry, to Navigation and Nautical 
Astronomy. 

It will occupy the student, properly acqtiainted with trigo- 
nometry, but a few days. Should it be omitted. Part V. will 
be found to contain a few additional general formulae, neces^ 
sary for the study of Analytical Geometry. 



PART IV. 



APPLICATION OF PLANE AND SPHERICAL TRIGO- 
NOMETRY TO THE PRINCIPLES OF NAVIGA- 
TION AND NAUTICAL ASTRONOMY. 

95. HAviNa in the preceding parts of the present treatise 
pretty fully explained and illustrated the principles of plane 
and spherical trigonometry, we shall now, for the purpose of 
showing the practical utility of these principles, apply them 
to the solution of one of the most important mathematical 
problems that ever has engaged the attention of man, viz. to 
determine the place of a ship at sea- 

When a ship sails from any known place, and a correct ac- 
count is kept of her various directions, and rates of sailing, 
her situation at any time may be readily ascertained by the 
rules of plane trigonometry, and the solution of the problem 
from these data belongs to Navigation. 

But it is impossible to measure a ship's course and the dis> 
tance sailed exactly ; so that after a long passage it would be 
unsafe to compute the place of the ship from the ship^s 
^ekoning. In such cases, therefore, the solution must 
I effected from other data, independent of the ship's account; 
iese are furnished by astronomical observation, and the com- 
putation is performed by the rules of spherical trigonometry ; 
the problem then becomes one of Nautical Astronomy. We 
•hall devote a distinct chapter to each of these important 
branches. 



PRINCIPLES OP NAVIGATION. 18B 



CHAPTER I. 
THE PRINCIPLES OF NAVIGATION. 

Dejmitions. 

96. 1. The earth is very nearly spherical. For the purpo"'^*' 
of Navigation it may be considered as perfectly so. It revolves 
round one of its diameters, called its axis, in about twenty- 
four hours. This rotation is from the west towards the east, 
causing the heavenly bodies to have an apparent motion firom 
the east towards the west. 

2. The great circle, whose poles are the extremities of the 
axis, is called the equator. The poles of the equator are 
called also the poles of the earth ; the one being the north 
pole, and the other the south pole. 

3. Every great circle which passes through the poles, and 
which, therefore, cuts the equator at right-angles, is called a 
meridian circle. Through every place on the surface of fhe 
earthsuch a great circle is supposed to be drawn ; it is the meri- 
dian of the place. It is expedientfor the purposes of Geography 
and Navigation to fix upon one of these meridians as a first 
meridian^ firom which the meridians of other places are 
measured. 

The English have fixed upon the meridian of Greenwich 
Observatory for the first meridian. 

4. The longitude of any place is the arc of the equator, 
intercepted between the meridian of that place and the first 
meridian ; the longitude, therefore, is the measure of the 
angle between the two meridians. The longitude is east 
west, according as the place is situated on the right or on the 
left of the first meridian, when we look towards the norti 
pole. 

6. The difference of longitude between two nUces : 
the arc of the equator, intercepted between the 

20 



106 APPLlCATfON OF TRIGONOHETRYr 

those places, or the measure of the angle which they include ; 
hence, when the longitudes of the places are of the same 
denominartion, that is, either both east or both west, the dif- 
ference 19 found by subtracting' the one from the other ; but 
when they are of contrary denominations the difference 
is found by adding the one to the other. 

6^ The latitude of a place is its distance from the equator, 
measured on the meridian of the place. Latitude, therefore, 
is north or south, according to the pole towards which it is 
ineasured, and cannot exceed 90^. 

7. The small cireles drawn parallel to the equator, are 
ceRed parallels of latitude. The arcs of a meridian, inter- 
cepted between two such parallels, drawn through any two 
places, measures the difference of latitude of those places : 
when the latitudes are of the same denomination, the differ- 
ence of latitude is found by subtraction, but when the deno- 
minations are not the same, the difference of latitude is found 
by addition, like difference of longitude. 

8. The horizon of anyplace is an imaginary plane, conceiyed 
to touch the surliBU^e of the earth at that place, and to be extend- 
ikI to the hpsivens ; such a plane is called the sensible hcrizonj 
and one paraUe]. to it, but passing through the earth's centre, is 
the rational horizon of the place. A line drawn across the 
horizon and throiigh the place, in the plane of its meridian, 
is called a north and south line ; the horizontal line through 
the same point, and perpendicular to this, is the east and 
west line. Besides the North, South, East and West, points 
thui? ^larked on the boundary of the horizon, this boundary 
Is qonceived to be subdivided into other intermediate points, 
porresponding to the divisions in the circle on the next page, 

9. The course of a ship is the angle which her track makes 
lyith the meridians; so long as this angle remained the same, 
if the meridians were all parallel, the path of the ship would 
be a straight line ; bqt as the successive ones bend towards 
that from which the ship sets out as you approach the poles, 
the direction of her path is continually changing, find she 



VBltTCirLSfl OF NAVtOATltftr^ 



tar 



moves in a curve, called the rhumb tine, or lomdrtmUe eurwe. 
The magnitude of the angle or the coune is indicated by 
the mariner's eampass. 

10. The mariner't compass consists of a circolar card, 
whose circumference is divided into thirty-two equal parts, 
called paints, and each of these is subdivided into lour eqnal 
parts, ca^ed quarter points ; aerosG this catd, and fasteded to 
it, so that th^ move together, is fixed a slender bar of mag- 
netised sted, called the needle ; the tapering extremities of 
which point to two diametricaUf opposite divisions of the 
card. These opposite divisions are marked N. and S., cor- 
vesponding to the 7u>rlh and smith poles, or ends, of the n 
netized bar. The 
diameter W.E., at 
rJg;ht angles to the 
diameter N, S., 
pointout the west 
and east points; 
these four are 
called the cardi- 
nal points, and the 
others are marked 
as in the subjoin- 
ed diagram. 

Thus one point 

from the north 

towards the east, is n»rtk by east ; two points, north, north- 
east ; three points, north-east by north; and so on, Each 
quadrant contains eight points, so that a point is 90°-»-8 = 
110 15'. (See Table of Rhumbs, Table V. at the end.) 

The caPd thus furnished being now suspended horizontally, 
so as to move freely and allow the needle attached to it, to 
settle itself will point out the four cardinal points of the 
horizon, as also the several intermediate points, provided only 
that it is the property of the magnetic needle to pomt due 
north and south. Such, however, is not strictly the case, as 
the needle is found, from accurate obaarationb, to deviate 




138 APPLICATION OF TRIGONOMETRY. 

from this position, and at some places very considerably, and 
this deviation is itself subject to variation. But the true 
direction of the compass, or the angle it makes at any place 
with a liiie pointing duly north and south, may be ascertained 
at any time by astronomical observations, and thus the devia- 
tion of the compass points, from the corresponding points of 
the horizon, may always be found and allowed for. 

The compass is so placed on ship-board that the vertical 
plane, cutting the ship from stem to stern, may pass through 
the centre of the card, so that that point of the compass which 
is directed to the ship's head shows the compass-course^ and 
the proper correction for variation being applied, the tTue 
course will be obtained. 

11. A ship's rcUe of sail- 
ing is determined by means 
of im instrument, called the 
leg, and an attached line, 
called the logAine. The 
log is a piece pf wood, form- 
ing the sector of a circle, 
and its rim is so loaded with 
lead, that when heaved into 
the sea, it assumes a vertical 
position, with its centre barely above the water. The log- 
line is so attached as to keep the face of the log towards the 
ship, that it may offer the greater resistance to being dragged 
after the ship by the log-line, as it unwinds from a reel on 
board, by the advancing motion of the ship. The length of 
line thus unwound in half a minute^ gives the rate of sailing. 
For convenience, the log-line is divided into equal parts, 
called knots^ of which each measures the 120th of a nautical 
r geographical mile* and as half a minute is the 120th of 
1 hour, it follows that the number of knots, and parts of a 
^ot, run in half a minute expresses the number of miles, and 
parts of a mile, run in an hour, at the same rate of sailing. 

* The geographical mile ia one minute of the earth'a circomfsrence. Taking 

• diameter at 7916 English miles, the geographical mile will be abont 6079 feet. 




PRIHCIPLBS OF HATIOATION. 



On Plane iSailing> 




97. Let the annexed 

diagram represent a 

portion of Uie earth's 

surface, p being the 

pole, and c « the 

equator. Let a b be 

any rhumb line, or 

track described by a 

ship in sailing on a 

single course irom a 

to B. Conceive the path of (he ship to be divided into portions 
A&, be, cd, &c., so small that each may differ inaensibly from 

a straight line, and draw meridians through these several 

divisions, as also the parallels of latitude bb', txf, ddf, &c.; we 

shall thus have a series of triangles described on the surface 
of the globe, but so small that each may be considered as a 
plane triangle. These triangles are all similar, for the angles 
at b', €, d', &c., are right-angles, and the ship's path cuts all 
the meridians at equal angles ; hence (Qeom., Prop. 18, B. 4,) 

^b : Ab' .:bc:he' : '. cd: cd', dec. 
therefore, (Geom., Prop. 6, B. 2,) 

Aft : a6' :: a6 + 6c + cd + dfcc, : a6' + Ac" + cd' + &c. 
But ib + be -\- cd + &«., is the vhole distance sailed, and 
Aft' + ftc' + ceC + &c. = ab', is the difference of latitude 
between a and b ; consequently, if a right angled triangle a 
bb', similar to the small triangle Aftft', be constructed, that is, 
one in which the angle a is equal to 
the course, and if the hypothenuse a b 
represent the distance sailed, the side 
ab' will represent the difierence of lati- 
tude. Moreover the other side bb', or 
that opposite to the course, will repre- 
sent the sum b'b -\- cc + d'd -f &c. of 
all the minute departures which the 
ship makes from the successive meri- 
dians which it crosses ; for as the tri- 




130 APPLICATION OF TRIOONOMfiTRT. 

angle abb', in this last diagram, is similar to the small triangle 
Abb'f in the former, we have 

a6 : &6' : : AB r bb' (1) ; 

but in the first figure we have 

Aft : bb' : ibcitc' it ed: dd^ &c. 

.-. a6 : 66' : : a6+6c'+ cd + &c. : 65'+ 00*+ cW+<kc (2) ; 

consequently, since the three first terms of (1) are respectively 
equal to those of (2), the fourth term bb', of (1), must be equal 
to the fourth term, 66' + ec* + rfd' + of (2), &c. This last 
quantity is called the departure of the ship in sailing firom a 
to b. It follows, therefore, that the distance sailed, the dif- 
ference of latitude made, and the departure, are correctly 
represented by the kypothenuse and sides of a right angled 
plane triangle, in which the angle opposite the departure is 
the course, so that when any two of these four things are given, 
the others may be found simply by the resolution of a 
right angled plane triangle ; as fiir, therefore, as these parti- 
culars are concerned, the results are the same as if the ship 
were sailing on a plane surface, the meridians being parallel 
straight Unes, and the parallels of latitude cutting them at 
right angles ; and hence that part of Navigation in which only 
distance sailed, departure, difierence of latitude, and course 
are considered, is called Plane sailing, 

EXAMPLES. 

1. A ship from latitude 47^ SO' N. has sailed S. W. by S. 
98 miles. What latitude is she in, and what departure has 

she made? 

c 

Let c be the place sailed from, cb the 
meridian, the angle c = -3 points =^ 
33^ 46', and ca = 98 miles, the dis- 
tance sailed ; then cb will be the differ- 
ence of latitude, and ba the departure, 
rice by the formulae for the solution of 
it angled triangles, 




PRINCIPLES OF NAVIGATION. 



131 



Asrad. ... 10. 
: Distance 98 1.991226 
: : COS. course 33Q45' 9.919846 

: Diff. of lat. 81.48 1.911072 



As rad. . . 

: Dist. . . 

: : sin. course 



10. 

1.991226 
9.744739 



Departure 54.46 1.736966 



Latitude left 47° 30' N. 

Diff. oflat.=81.48minute3 =l 22 S. Dep.=64.46 miles W. 

Latitude in 46 8 N. 

2. A ship sails for 24 hours on a direct course, from lat. 

38^ 32^ N., till she arrives at lat. 36^ 66' N. ; the course is 

between the S.'and E.,and the rate 6^ miles an hour. Required 

the course, distance, and departure. 

Lat. left 380 32' N. 24 x 61 = 132 miles, the distance. 
Latin 36 66 N. 

Diff. 1 36 = 96 miles 

As Dist. 132 2.120674 As rad . . . . 
Rad. ... 10. : Dist. . . . 

Diff. lat. 96 1.982271 : : sin. course . 



COS. course43o20' 9.861697 



10. 

2.120674 
9.836477 



: Dep. 90.68 . 1.967061 



Hence the course is S. 43^ 20' E., and the departure 90.68 
miles E. 

3. A ship sails from lat. 3<^ 62' S. to lat. 4^ 30' N., the course 
being N. W. by W. ^ W. ; required the distance and departure. 

Distance, 1066 miles ; Departure, 938.9 miles W. 

4. Two ports lie under the same meridian, one in latitude 
62° 30' N., and the other in latitude 47° 10' N. A ship from 
the southernmost sails due east, at the rate of 9 miles an hour, 
and two days after meets a sloop which had sailed from the 
northernmost port ; required the sloop's direct course and 
distance run. 

Course S. 63° 28' E., or S. E. J E. ; distance run 637.6 mf ^ 
8. If a ship from lat. 48^ 27' S. sail S. W. by W. 7 n 
^ hour, in what time will she arrive at the parallel of 60^ 

In 23/ 
6, If after a ship has sailed from lat. 40^ ' 



132^ APPLICATION OF TRIGONOMETRY. 

18' N., she be found 216 miles to the eastward of the port 
left ; required her course and distance sailed. 

Course N. 31® 11' E., distance 417.3 miles. 

Traverse Sailing. 

98. When a ship, in going from one place to another, sails 
on different courses, it is called traverse sailing ; and the 
determination of the single course and distance from the one 
place to the other is called working or compounding the tra- 
verse. To effect this, it is obviouidy merely necessary to find 
the difference of latitude, and departure, due to each distinct 
course, to take the aggregate of these for the whole difference 
of latitude and departure, and from these to find, as in last 
article, the single course and distance. It is usual in thus 
compounding courses to form a table consisting of six co- 
lumns, called a traverse tahle^ and in the first column to 
register the several component courses, and against them, in 
the second column, the proper distances ; the next two co- 
lumns, marked N. and S., ar to receive the several differ- 
ences of latitude, whether N. or S., due to each course, and 
distance, and the two remaining columns marked E. and W. 
are to receive, in like manner, the corresponding eastings 
and westings^ that is, the departures. When these several 
particulars are all inserted, the columns are added up, and 
the difference of the results of the N. and S. columns will be 
the required difference of latitude, and the difference of the 
results of the E. and W. columns will be the corresponding 
departure. (See page 1 34.) 

The columns appropriated to the differences of latitude and 
departures are usually filled up from a table already computed 
to every quarter point of the compass, and to all distances from 
one mile up to 100 or 120 ; so that, by entering this table with 
any given course and distance, the proper difference of lati- 
tude and departure is found by inspection. Most books on 

vigation and also surveying, contain a second and more 
irged traverse table, being computed to every course from 
larter of a degree up to forty-five degrees. This latter 



. PRINCIPLES OF NAVIGATION; 133 

table we have not thought it necessary to insert in our coUeo^ 
tion, but the former we have given, (Table IV.) 

This table shows, by inspection, the difference of latitude 
and departure due to any proposed course and distance. The 
course is found at the top or bottom of the page, and the dis- 
tance at the left or right of the half page. If the distance 
sailed be more than 120 miles, it will exceed the limits of the 
table ; but the difference of latitude and departure may ^ill 
be determined from it by this simple operation : divide the 
given distance by any number that will give a quotient ijot 
exceeding 120 ; enter the table with this quotient, and mul- 
tiply the corresponding diff. of lat. and dep. by the assumed 
divisor, and there will result the diff. of lat. and dep. due to 
the proposed distance. Or take any numbers whose sum is 
equal the given distance, the sum of their differences of lat 
and dep. will be the lat. and dep. of the given distance. These 
rules depend upon the principle that for the same course the 
differences of latitude and departure ate proportional to the 
distance run ; which will be evident if we recollect that dist.j 
diff. of lat. and dep. form a right angled triangle, and that two 
right angled triangles are similar when an acute angle of oile 
is equal to an acute angle in the other. 

But there is another mode of finding the direct course and 
distance, much practised by seamen, viz., by construction. 
To facilitate this construction the mariner^s scale is em- 
ployed, which is a two foot flat rule exhibiting several seakui 
on each side, by help of which and a pair of compasses the 
usual problems in sailing may be all solved. One of these 
scales is a scale of chords, commonly called a scale of rhumbs, 
being confined to every quarter point of the compass ; and 
another is a more enlarged scale of chords, being to every 
single degree. Both these scales are constructed in reference 
to the same common radius, so that the chords on the scale 
of rhumbs belong to that circle whose radius equals the chord 
of 60^ on the scale of chords ; and the method of laying down 
a traverse from these scales, and one of equal parts, and of 
thence measuring the equivalent single course, and distano 

81 



134 



APPLICATION OF TRIGONOMETRY. 



made good| will be at once nnderstood from the following 
examples. 

EXAMPLES. 

1. A ship sails from a place in lat. 24^ 32' N., and has run 
the following courses and distances, viz. 

Isty S. W. by W., distance 45 miles ; 2d, E. S. E., distance 
60 miles ; 3d, S. W., distance 30 miles ; 4th, S. E. by E., 
distance 60 miles ; 6th, S. W. by S., ^ W.. distance 63 miles : 
re4uired her present latitude, with the direct course and dis 
tance from the place left to the place arrived at. 

TVctverse Table. 



Ck)urses. 


Dist. 


Difference of lat. 


Departure. | 






N. 


S. 


E. 


W. 


S. W. by W. 


46 




26.0 




37.4 


E. S. E. 


60 




19.1 


46.2 




S.W. 


30 




21.2 




21.2 


S. E. by K 


60 




33.3 


49.9 




S. W. by S. JW. 


63 




60.6 




37.5 


\ 






149.2 


96.1 


96.1 



It appears from the results of this table that the difference 
of latitude made by the ship during the traverse is 149.2 S. 
= 29 29' 8. 

Lat. left 240 32' N. 

Diff.lat : 2 29 S. 



Lat. in 22 3 N. 

It appears also that the departures east are equal to the 
departures west, so that the ship has returned to the meridian 
she sailed from, consequently the direct course from the place 
left to that come to is due south, and the distance is equal to 
the difference of latitude, which is 149.2 miles. 



FRINCIPLEl OF NAVIGATIOK. 



135 



The construction of the traverse is as follows : 

With the chord of 60°, taken from 
the line of chords on the mariner's 
scale, describe the horizon circle, 
and draw the north and south line 
N. S. Prom the line of rhumbs 
take the chords of the several 
courses, and as these are all south- 
erly they must be laid off from the 
south point S, tliose which are west- 
erly to the left, and those which are 
easterly to the right, their extremi- 
ties being marked 1, 2, 3, &c., in the 
order of the courses. This done, 
lay off from any convenient scale of 
equal parts, and in the direction of 
a1 the distance ab sailed on the 
first course ; then in the direction 
parallel to a2, the distance bo sailed on the second course ; 
in the direction parallel to a3, the distance cd on the third 
course ; in the direction parallel to a4, the distance de on 
the fourth course ; and, lastly, in the direction parallel to aS, 
the distance ef on the fifth course ; then f will represent the 
plane of the ship at the end of the traverse ; fa, being applied 
to the scale of equal parts, will show the distance made good, 
and the chord of the arc included between this distance, and 
the meridian, being applied to the line of rhumbs, will show 
the direct course. In the present case the intercepted arc 
will be 0, showing that f is on the meridian of a. 

2. A ship from Cape Clear, in lat. 61^ 26' N., sails 1st, SJ3.R 
J E., 16 mUes ; 2d, E.S.E., 23 miles ; 3d, S.W. by W. ^ W., 
36 miles ; 4th, W. f N., 12 miles ; 5th, S.E. by E. J E., 41 
miles : required the distance made good, the direct course 
and the latitude in ? 




136 



APPLICATION OF TRIGOMOMETRT. 



Traverse Table. 



Couises. 


Dist. Difierenceof Lat. 


Departure. 


S.S.E.^E. 

fi.S.E. 

8.W. by W. I W. 

W. iN. 

S.E. hjE. \ E. 


16 
23 
36 
12 
41 


N. 
1.8 


14.5 

8.8 

17.0 

21.1 


E. 
6.8 
21.3 

35.2 


W. 

31.8 
11.9 




« 


1.8 


61.4 

1.8 


63.3 
43.7 


43.7 




69.6 


19.6 



^at. left 

Piff. lat. SQ'.em 

Lat^ in t 



51° 25' N. 
I S. 



50 25 N, 



TheQ by the formulae for the solution of right-angled plane 
triangles. 

A8diff.lat. 69.6 1.775246 
I rad. - 10. 

: : departure 19.6 1,292256 



^ tan, course X80il2' 9.517010 



As sin. course 


- 9.494621 


; departure - 


1.292256 


: ! rad. 


- 10. 



: distance 62.74 1.797635 



therefore, as the difference of latitude is south, and the de^ 
parture east, the direct course is S. 18° 12' E., and the dis- 
tance made good 62.74 miles. 

To construct this traverse, describe, as before, the horizon 
circle, with a radius equal to the chord of 60°, and taking 
from the line of rhumbs the chord of the first course, 2J 
points, apply it from S. to 1, to the right of S.N., as this 
course is south-easterly ; apply, in like manner, the chord of 
*hQ second course, six points from S. to 2, also to the right of 
meridian line ; apply the chord of the third course, 6^ 
Its from S. to 3, to the left of the meridian, the chord of 



PRINCIPLES OF NAVIGATION. 



137 



the fourth course, 7^ from N. to 4, to the left of N. S., this 
course being north-west- 
erly, and, lastly, apply 
the chord of the fifth 
course, 5 J points, from S. 
to 6, to the right of S.N. 
In the direction a1, lay 
off the distance ab = 16 
miles from a scale of 
equal parts ; in the direc- 
tion parallel to A2,Iay off 
the distance bc = 23 
miles; in the direction 
parallel to a3, lay off cd 
= 36 ; in the direction 
parallel to a4, lay off de 
= 12 miles ; and, lastly, 
in the direction parallel 
to a5, lay off ep = 41 ; 
then F will be the place 
of the ship at the end of 
the traverse ; consequently, af will be the distance made 
good, and the angle fas the direct course ; applying, there- 
fore, the distance af to the scale of equal parts, we shall 
find it reach from to 62| ; and applying the distance Sa to 
the line of chords, we shall find it reach from to 18^. 

3. A ship from lat. 28^ 32' N., has run the following 
courses, viz. 1st, N.W. by N., 20 miles ; 2d, S.W., 40 miles ; 
3d, N.E. by B., 60 miles ; 4th, S.E. 55 miles ; 5th, W. by S., 
41 miles; 6th, E.N.E., 66 miles. Required her present 
latitude, the distance made good, and the direct course from 
the place left to that come to. 

The direct course is due east, and distance 70.2 miles, the 
ship being in the same latitude at the end as at the beginnir^* 
of the traverse. 

4. A ship from lat. 41^12' N., tails S.W. by W:, 21 mil 
S.W. i S. 31 miles ; W.S.W. ^ S., 16 miles ; S. | E.. !« mil 




138 APPLICATION OF TRIGONOMETRY. 

S. W. ^ W., 14 miles ; and W. ^ N., 30 miles : required the lati- 
tude of the place arrived at, and the direct course and dis- 
tance to it. 

Lat. 40O 6' N.; course S. 52° 49' W.; distance 111.7 miles. 

6. A ship runs the following courses, viz. 

1st, S.E., 40 miles; 2d, N.E., 28 miles; 3d, S.W. by W., 
62 miles ; 4th, N.W. by W., 30 miles ; 6tb, S.S.E., 36 miles ; 
6th, S.E. by E., 68 miles : required the direct course and dis- 
tance made good. Direct course S. 26<^ 69' B., or S.S.E J E. 
nearly ; distance 95.87 miles. 

These examples will, perhaps, suiSce to illustrate the prin- 
ciples of plane sailing, in which, course, distance, difference 
of latitude, and departure, are the only things concerned. 
The determination of the difference of longitude made on any 
course which is the distance between the meridians measured 
on the equator, cannot be effected by these principles, for this 
element is not the same as if the meridians were all parallel 
to each other, as is the case with the other elements. The 
finding of the difference of longitude is the easiest when the 
ship sails due east or due west, that is upon a parallel of lati- 
tude ; this is called parallel sailing. 

Parallel Sailing. 

99. The theory of parallel sailing is comprehended in the 
following proposition, viz : 

The cosine of the latitude of the parallel is to the distance 
run as the radius to the difference of longitude. This may 
be demonstrated as follows 

Let laH represent the 
equator, and bda any paral- 
lel of latitude ; ci will be the 
radius of the equator, and cb 
tiiie radius of the parallel. 
Let BD be the distance sail- 
A, then the difference of 
jitude will be measured 
the arc la of the equa- 




PRINCIPLES OF NAVIGATION. 139 

tor, and since (Geom., Prop. 11, Cor. B. 5,) similar arcs are to 
each other as the radii of the circles to which they belong, 
we have 

CB : CI : : dist. bd : diff. long. la. 

But CB is the cosine of the latitude ib to the radius ci, 
that is, 

CB : ci : : cos lat : a 

The first two terms of these proportions being the same, 
the last are proportional, and we have 

COS. lat. : Rad. : : distance : diff. long. . . . (1) 
Corollary : hence if the distance between any two meridians, 
measured on a parallel in latitude l be d, and the distance of 
the same meridians, measured on a parallel, in latitude l' be 
d', we shall have (Geom., Prop. 11, Cor. B. 5,) 

cos L : D : : cos l' : d' . . . (2) 
By referring to proportion (1) it will 
be seen that if one of the legs of a 
right-angled triangle represent the dis- 
tance nln on any parallel, and the ad- 
jacent acute angle be equal to the de- 
grees of lat. of that parallel, then the 
hypothenuse will represent the differ- 
ence of longitude, since this hjrpothenuse will be determined 
by that proportion. It follows, therefore, that any problem 
in parallel sailing, may be solved by the traverse table, com- 
puted to degrees, as a simple case of plane sailing ; for by 
considering the latitude as the course, and the distance as the 
difference of latitude, the corresponding distance in the table 
will express the difEerence of longitude. 

EXAMPLES. 

1. A ship from latitude 53° 56' N., longitude 10° 18' E., 
has sailed due west, 236 miles : required her present longi- 
tude. 




140 APPLICATION OF TRIGONOMETRY. 



By the rule 
Ascoslat 630 56' - 
: radius ... 
: : distance 236 


* 

10° 
6 


• 

- 9.769913 
- 10. 
' 2.372912 


:diff.long. 400.8 


- 2.602999 


Long, left 
Diflf. long. = -^ degrees = 


18' E. 
40 W. 


Long, in 


3 


38 E. 



2. If a ship sail E. 126 miles, from the North Cape, in lat. 
71° 10' N., and then due N., till she reaches lat. 73° 26' N.; 
how £sur must she sail W. to reach the meridian of the North 
Cape? 

Here the ship sails on tv^o parallels of latitude, first on the 
parallel of 71° 10', and then on the parallel of f 3° 26', and 
makes the same difference of longitude on each parallel. 
Hence, by the corollary. 

As COS. lat. 71° 10' arith. comp. 0.491044 
: distance 126 . . 2.100371 
: : COS. lat. 73 26 . . 9.455044 



: distance 11L3 . . 2.046459 



3. A ship in latitude 32^ N* sails due east, till her differ- 
ence of longitude is 384 miles ; required the distance run. 

325*6 miles. 
4 If two ships in latitude 44^ 30' N., distant from each 
other 216 miles, should both sail directly south till their dis- 
tance is 256 miles, what latitude would they arrive at ? 

32oi7'S. 
5. Two ships in the parallel of 47° 54' N., have 9° 35' dif- 
ference of longitude, and they both sail directly south, a dis- 
tance of 836 miles : required their distance from each other 
t the parallel left, and at Aat reached. 

385.5 miles, and 479.9 miles. 



fniilClPLEa OF NAVtOATlOS; 141 

Middle Latitude Sailing. 

100, Having seen how the longitude which a ship makes 
^hen sailing on a parallel of latitude may be determined, we 
come now to examine the more general problem, viz. to find 
the longitude a ship makes when sailing upon any oblique 
rhumb. 

There are two methods of solving this problem, the one by 
what is called middle latitude sailing, and the other by 
Mercator's sailing. The first of these methods is confined 
in its application, and is moreover somewhat inaccurate even 
where applicable ; the second ia perfectly general, and rigor- 
ously true ; but still there are cases in which it is advisable 
to employ the method of middle latitude sailing, in preference 
to that of Mercator's sailing ; itis, therefore, proper that mid- 
dle latitude sailing should be explained, especially sinte, by 
means of a correction to be hereafter noticed, the iisual inaccu- 
lacy of this method may be rectified. 

Middle latitude sailing proceeds on the supposition that 
thedepartureorsum 

Of all the meridional ^^?=:r^ 

< distances b'b, c'c,d'd, 
Slc. from a to b, is 
equal to the distance 
M'Hof the meridians 
of A and B, measured 
on the middle pa- 
rallel of latitude be- 
tween A and B. 

This supposition becomes very inaccurate when the course 
is small, and the distance run great ; for it is plain that th^ 
middle latitude distance will receive a much greater accession 
than the departure, if the track a b cuts the successive me- 
ridifms at a very small angle. 

The principal approaches nearer to accuracy as the angle 
a of the course increases, because' then as but httle advance 
is made in latitude, the several component departures lie mote 
in the immediate vicinity of the middle latitude parallel. But 




142 APPLICATION OF TRIGONOMETRY^ 

Still, as in very high latitudes, a small advance in latitn^e 
makes a considerable difference in meridional distance, this 
principle is not to be recommended in such latitudes if much 
accuracy is required. 

By means, however, of a Small table of corrections, con^ 
structed by Mr, W&rkman^ the imperfections of the middle 
latitude method may be removed, and the results of it ren- 
dered in all cases accurate. This table we have given at the 
end of the present volume^ 

The rules for middle latitude sailing may be thus deduced. 

It has been seen at (Art. 97) that the difference of latitude, 
departure, and distance, sailed on any j^ 
oblique rhumb, will be all accurately repre- 
sented by thfe sides ab', b'b, ab, of a plane 
triangle. Now, by the present hypothesis, 
the departure b'b is equal to the middle 
latitude distance between the meridians of 
the places sailed from, and arrived at, so 
that the diflference of longitude of the two 
places of the ship is the same as if it had 
sailed the distance b'b, on the middle lati- "^ 
tude parallel ; the determination of this difference of longitude 
is, therefore^ reduced to a case of parallel sailing, for bb', now 
representing the distance on the parallel, and an angle a' bb' 
being made equal to the latitude of that parallel, we shall 
have the difference of longitude, represented by the hypothe- 
nuse a'b. We thus have the following theorems, viz., in the 
triangle a^b'b, 

cos a'bb' : bb' : : radius : ba' 
that is, 

I. Cos. mid. lat. : departure : : radius : diflf. of long. 

In the triangle a'ba, 

sin. A : AB : : sin. a : a'b ; 

that is, 

II. Cos. mid. lat. : distance : : sin. course : diff. long. 
In the triangle abb', we have the proportion, (Art. 41) 

R : tan a : : ab' : bb' 




PRINCIPLES OP NAVIGATION. 143 

comparing this with the first proportion above, observing that 

the extremes of this are the means of that, we have 

ab' : a'b : : cos. a'bb' : tan. a ; 
that is, 

III. Diff. lat. : diff. long. : : cos. mid. iat. : tan. course. 

These three proportions comprise the theory of middle 
latitude sailing, and when to the middle latitude the proper 
correction, taken from Mr Workman's table, is added, these 
theorems will be rendered strictly accurate. 

This is Table YI ; the middle latitude is to be found in the 
first column to the left ; in a horizontal line with which, and 
under the given difference of latitude, is inserted the proper 
correction to be added to the middle latitude to obtain the 
latitude in which the oaeridian distance is accurately equal to 
the departure. The formula for constmctixig this table ii 
obtained as follows :* 

Let 

d =: proper diff. of lat. 

D = meridional diff. of lat 
m = middle latitude. 
M = w» + correction^ 
L = diff. of longitude. 

Then, (Art. 100, Form III.) 

cos M X L 

tan course = -^ 

But, (Art. 101, Rule 1.) 

rad. X L 
tan course = — 

D 

COS. M X L rad. x l rad. d 

= .-. COS. M c:= 



4 



rad. d 
correction = cos.— ^ — r — — f» 



EXAMPLES. 

1. A ship, in latitude 51° 18' N., longitude 22^ 6' W., i| 



»• m 



• The iivrestlgation of this formuisi nhwk^ be poBt^oned till «fter reading 
tbe next article, and may be omitted en^rely. 



14i APPLICATION OP TRIGONOME'TRY. 

bound to a place in the S. E. quarter, 1024 miles distant, and 
in lat. 37^ N. : what is her direct course and distance, as alse 
^he difference of longitude between the two places ? 

Lat. from 61° 18' N. ) ^ ^, . . oo. .o, 

Lat. to 37 ON. Sum of lat^ides . «8o 18' 

^ Mid. lat. .... 44 9 

Difflat. 14 18 = 868 miles. 



For the course. 
As distance 1024 3.010300 
: radius 10. 

diff. lat. 858 2.933487 



ft • 



: : COS. course 33° 5' 9.923187 



For the diff. long. 

cos. mid. lat. 44° 9' ar. comp. 0.144167 

: tan.course33 5 . . . 9.813899 

:: diff. lat. 868 ... . 2.933487 



: diff. long. 779 ... . 2.891553 



In this operation the middle latitude has not been corrected, 
so that the difference of longitude here determined is not 
without error. To find the proper correction, look for the 
given middle latitude, viz. 44° 9' in the table of corrections, 
the nearest to which we find to be 45^ ; against this and under 
14P diflF. of lat. we find 27', also under 15° we find 31', thp 
difference between the two being 4' ; hence corresponding to 
14P 18' the correction will be about 28'. Hence the corrected 
piddle latitude is 44° 37', therefore, 

pos. corrected mid. lat. 44^ 37' ar. comp. 0.147629 

: tan. course 33 5 . . . 9.813899 

:: diff. lat. 858 .... 2.933467 



diff. long, 785.3 .... 2.895015 



therefore, the error in the former result is about 6i miles. 

2. A ship sails in the N.W. quarter, 248 miles, till her de- 
parture is 135 miles, and her difference of longitude 310 
miles : required her course, the latitude left, and the latitude 
pome to. 

Course N. 32° 59' W. ; lat. left 62° 27' N. ; lat. in 65° 55' N. 

3. A ship, from latitude 37° N., longitude 9° 2' W., having 
sailed between the N. and W., 1027 miles, reckons that she has 
made 564 miles of departure ; what was her direct course, 
and the latitude and longiti|de reached ? 

Course N. 33° l9' W. or N» W. by N. nearly ; lat. 51° 18' 
N. : long. 22° 8' W. 



PRINCIPLES OF NAVIGATION. 



14S 



4. Required the course and distance from the east point of 
St. Michael's, lat. 37^ 48' N., long. 25^ 13' W., to the Start 
Point, lat. &00 13' N., long..3o 38' W., the middle latitude being 
corrected by Workman's Table. 

Coarse N. SP 11' E. ; distance 1189 miles. 

Mercator^s Sailing. 

101. It has been already seen that when a ship sails on any 
oblique rhumb, the difference of latitude, the departure, and the 
distance run, are truly represented by the sides of a right-an» 
gled plane triangle. The departure b'b 
represents the sum of all the very small 
meridian distances, or elementary de- 
partures, b'b, cc, &c. in the diagram, at 
Art. 100, the difference of latitude ab' 
represents the sum of all the correspond- 
ing small differences in the figure refer- 
red to ; and the distance ab, the sum of 
all the distances to which these several 
(departures and differences belong, and 
each of these elements is supposed 
to be taken so excessively small as to form on the sphere 
a series of triangles, differing insensibly from plane triangles. 

Let Ab'b in the annexed diagram represent one of these 
elementary triangles, b'b will be one of the elements ofthede-. 
parture, and a6', the corresponding difference in latitude ; 
and as b'b is a small portion of a parallel of latitude, it will 
be to a similar portion of the equator, or of the meridian, as 
the cosirie of its latitude to radius (Art. 99). This similar 
portion of the equator, or of the meridian, being the differ^ 
ence of longitude between b' and 6. Suppose now the dis- 
tance a6 prolonged to p, till the departure p'p is equal to the 
difference of longitude of 6', and 6, then b'b will be to p^p as 
the cosine of the latitude of b'b to the radius ; but b'b : p 
: : Ab' : Ap' ; hence the proper difference of latitude a6' is 
the increased difference a/?' as the cpsine of the latitude of i 




146 APPLICATION OP TRiaONOMETRY. 

to the radius. Calling, therefore, the proper difference of 

latitude d, the increased difference of latitude d, the latitude 

of b'b, I, and the radius 1, which it is in the table of natural 

sines, we have 

d 1 

D= ; =rf sec. /since sec == — . . . (Art. 33) 

cos. I cos ^ ^ 

The ship, therefore, having made the small departure b'b, and 
the difference of latitude a6', must continue her course till 
the difference of latitude becomes d, in order that her depar- 
ture may become equal to the difference of longitude corres- 
ponding to b'b. Conceiving all the elementary distances to 
be in this manner increased, the sum of all the corresponding 
increased departures will necessarily be the whole difference 
of longitude made by the ship during the course ; to repre- 
sent, therefore, the difference of longitude due to the depar- 
ture b'b, and difference of latitude ab', we must prolong ab' 
till Ac' is equal to the sum of all the elementary differences 
increased as above, and the departure c'c, due to this differ- 
ence of latitude, will represent the difference of longitude 
actually made in sailing from a to b. The determination of 
Ac' requires the previous determination of all its elementary 
parts ; if d be talien equal to 1', each of these parts will be 
expressed by d = 1' sec. Z, or d = sec. Z, it being understood 
that sec. I expresses so many minutes or geographic miles, 
from which equation the values of d, corresponding to every 
minute of Z, from the equator to the pole, may be calculated ; 
«uid by the continued addition of these there will be obtained, 
in succession, the values of the increased latitude corresponds 
to 1', 2', 3', &c. of proper latitude ; these values are called the 
iiwrMZionaZpar^^, corresponding to the several proper latitudes, 
.4in4 when registered in a table, form a table of meridional 
parts, given in all books on Navigation. 

The foUowiag may serve as a specimen of the meinner in 

which such a table maybe constructed, and, indeed, of the maur 

in which th,e first table of meridional parts was actually 

led by Mr, Wright, the proposer of this ingenious an4 

table method. 



PRINCIPLES OP NAVIGATION. 147 

Mer. pts. of 1'= nat. sec. 1'. 
Mer. pts. of 2'= nat. sec. 1' + nat. sec. 2'. 
Mer. pts, of 3'= nat. sec. 1' + nat. sec. 2' -|- nat. sec. 3'. 
Mer. pts. of 3'= nat. sec. 1' + nat. sec. 2' + nat. sec. 3'+nat, 
sec. 4', 

(fcc. &c. 

Hence, by means of a table of natural secants, we have 

Nat. sees. Mer. parts. 

Mer. pts. of 1^= . . 1.000000 = 1.0000000 

Mer. pts. of 2'== 1.0000000 + 1.000000 = 2.0000002 
Mer. pts. of 3'= 2.0000002 + 1.0000004 = 3.0000006 
Mer. pts. of 4'= 3.0000006 + 1.0000007 = 4.0000013 
&c. (fee. 

There are other methods of construction, but this is the 
most simple and obvious. The meridional parts, thus de- 
termined, are all expressed in geographical miles, because in 
the general expression d = 1' sec. /, 1' is a geographical mile. 
Having thus formed a table of meridional parts, (Table YIL 
at the end,) if we enter it with the latitudes sailed from, and 
come to, and take the difference of the corresponding parts in 
the table, the remainder will be the meridional difference of 
latitude, or the line ac' in the preceding diagram, and the dif- 
ference of longitude c'c will then be obtained by this propor- 
tion, viz. 

1. As radius is to the tangent of the course, so is the 
meridional difference of latitude to the difference of longu 
tude; or if the departure be given instead of the course, then 
the proportion will be 

2. As the proper difference of latitude is to the depar* 
ture, so is the meridional difference of latitude to the tan' 
gent of the course. Other proportions immediately suggest 
themselves from the preceding figure. 

102. As an example of Mercator's, or more properly of 
Wright's, sailing, let us take the following : 

1. Required the course and distance from the east poi© 
St. Michael's to the Start Point. 



148 



APPLICATION OP TRIGONOMETRY. 



Start lat. 60° 13' N. Mer. pts. 3495 long. 3° 38' W. 

St. Michael's lat. 37 48 N. Mer. pts. 2453 long. 25 13 W. 



12 26 

60 



Mer.diff.l042diff.long.21 35 W« 
60 



Proper diff. lat. 745 miles. 



1295 mileiS. 



For the course. 
AsMer.diff.lat.1042 3.017867 
: radium . . 10. 
i: diff. long. 1295 3.112270 



I tancourse51oil'E.10.094403 



Pot the distance; 
As COS. course 9.797150 

: prop. diff. lat. 2.872156 



: : rad. 



10. 



: distance 1189 3.076006 



A ship iSails from lat. 37° N. long. 22° 56' W., on the course 
N. 330 i9' E., till she arrives at 5P 18' N. : required the dis- 
tance sailed, and the longitude arrived at. 

Dist. 1027 miles ; long. 9° 45' W. 

We shall here terminate the present chapter on the prin- 
ciples of Navigation, having now discussed the several cases 
of sailing which actually occur in practice. But the student 
who is desirous of prosecuting his inquiries on this very 
important branch of practical science to greater extent, will, 
of course, consult works expressly devoted to the subject. 
Of these the most elaborate in our own language is the 
valuable " Elements" of Robertson, in two octavo volumes. 
The Treatise of Mr. Riddle is also an excellent work) 
abounding with practical examples very accurately solved, 
and upon the whole better adapted to modern practice, 
as well as more compendious, than Robertson's. Mr, 
Norie's Navigation, is also a good practical book. Dr^ 
Bowditch's work is the principal American one, and is, 
oerhaps, the best practical book on the subject extant. 



NAUTICAL ASTRONOMY. 149 



CHAPTER II. 

APPLICATION OF SPHERICAL TRIGONOMETRiT 

TO NAUTICAL ASTRONOMY. 

103. In our chapter on Navigation we have laid dowii 
several methods of determining the place of a ship at sea^ 
by help of the account kept on board of its progress through 
the water, that is, of the course and distance sailed ; and, if 
confidence could be placed in this Account, even when kept 
with the utmost care, the art of Navigation would be perfect 
Such perfection, however, it is hopeless to expect ; for it doed 
hot seem possible to measure, with strict accuracy, either a 
ship's rate or the direction in which she moves, both of 
which may indeed be continually varying. In order, there^ 
fore, to determine the place of a ship at sea, with that accu- 
racy which the safety of navigation requires, it is absolutely 
necessary that we be furnished with methods entirely inde- 
pendent of the dead reckoning, and these methods it is the 
business of Nautical Astronbmy td teach. 

" It must not, however, be understood that the dead reckon- 
ihg is without its value ; on the contrary, when combined 
with astronomical observations, it is of considerable utility 
ih detecting the existence and velocity of currents, and is in- 
dispensably necessary to fill up the short intervals which may 
Occur in Unfavorable weather between celestial observations; 
But the too general practice of relying exdusively upon it 
cannot be sufficiently deprecated, and numerous instances 
rtiight be adduced of the fatal consequences of this reliance, 
in the loss of vessels, from errors of such magnitude that 
they might have been detected by the most superficial know- 
ledge of nautical astronomy, and the aid of even a goa 
common watch." (Capt. Kater^s Nautical Astronomy in tl 
Ency. Met.) 



2S 



160 APPLICATION OF TRIGONOME^Rl^, 

t 

Definitions. 

104. For the purpose of measuring the angular distance* 
of the heavenly bodies from each other, and from the horizon, 
it is convenient to suppose them all situated as they really 
appear to an observer on the earth, viz. in a spherical con- 
cave surrounding our earth; and concentric with it. This 
imagmary concave is called the celestial sphere, or the ap- 
parent heavens; in it all the apparent motions of the heaven- 
ly bodies are, for the convenience of trigonometrical appli- 
cation, supposed actually to take place ; and the entire celes- 
tiat sphere to revolve daily round the earth as if this were at 
rest in its centre. Ail this is allowable, beeatisethe applica- 
tions of which we speak are not affected by the inquiry, 
whether the motions which the heavenly bodies present to 
an observer on the earth are really as they appear or not. 

At Art. 79, we defined several lines, some of which, geo- 
gi^aphers have found it convenient to consider as described on 
the surface of the earth ; but which astronomers extend t€> 
the heavens. 

We give here some additional matter on the same sub^ 

ject. 

.^ The position of any point on the celestial sphere, like the 
position of a point on the terrestrial sphere, is marked out, as 
we have said, by its latitude and longitude. On the celestial 
sphere the circle of longitude is the ecliptic ; and perpendi- 
culars to this, passing, therefore, through the poles of the 
ecliptic, are the circles of celestial latitude ; the point from 
which longitude is measured is the vernal equinoctial points 
CJommencing at this point, too, the ecliptic is divided into 
twelve parts, called signs ; a sign is, therefore, 30°. The 
twelve signs are named, and symbolically expressed, as fol- 
lows: 

1. T Aries. 

2. b Taurus. 

3. n Gemini. 



4. 2o Cancer. (7. -^o> Libra. |10, V3 Capricornus. 



5. £i Leo. 

6. HK Virgo. 



8. itl Scorpio, jll. ^ Aquarius. 

9. / Sagittarius.' 12. ^ Pisces. 



The fir^t six of these signs are on the north of the equi- 



MAUTICAL ASTROXOMY. 151 

noctial, the others on the south, and the rernal equinoctial 
point is called the first point of Aries. The longitude is 
measured from tliis point in but one direction, viz. in the 
order of the signs. 

Parallels of latitude on the terrestrial sphere correspond to 
parallels of declination on the celestial. Of these, the two 
which are 23° 28' from the equinoctial, one on each side, and 
which therefore touch the ecliptic in the first points of Can- 
cer and Capricorn, are called the tropics of Cancer and of 
Capricorn. These first points of Cancer and Capricorn are 
respectively called the summer and winter solstice ; because 
for a day or two before and after the sun enters them he ap- 
pears to be stationary, and the days to be of equal length, so 
slowly does his declination at those times change, for his mo- 
tion is obviously very nearly parallel to the equinoctial. The 
meridian, through the solstitial points, is called the solstitial 
colure, and that through the equinoctial points, the equinocr 
Hal colure. 

Having described the principal circles and points of the 
celestial sphere which are considered as permanent, or which 
4o not alter with the situation of the observer on the earth, 
we come now to describe those which change with his place. 
The principal of these is the horizon^ which has been defined 
already, (Art. 79,) and vertical circles^ which are perpendicu- 
lar to the horizon, and on which the altitudes of celestial 
objects are measured. 

These vertical circles all meet in two points diametrically 
opposite, viz., the poles of the horizon ; one of which is directly 
over the head of the observer, and called his zenith^ and the 
opposite one his nadir. That vertical one whicli passes 
through the east and west points of the horizon is called the 
prime vertical ; it necessarily intersects the meridian of the 
place (which passes through the north and south points) at 
right angles. 

The azimuth of a celestial object has been already defin 
to be an arc of the horizon, comprised between the meridi 
of the observer and the vertical circle through the oH 
hepce vertical circles are sometimes called azimuti 



162 APPLICATION OP TRIGONOMETRY. 

The amplitude of a celestial object is the arc of the hori-i 
zon, comprised between the east point and the point wjiere 
the object rises, or between the west point and that where it 
sets ; the one is called the rising amplitude, the other the 
getting amplitude. 

Qn the Corrections to be applied to the observed Altitudes 

of Celestial Objects, 

(105.) The tJ^ue altitude of a celestial object is alwajrs 
understood to mean its angular distance from the rational 
horizon of the observer. This is not obtained directly by ob- 
servation ; but is the result of certain corrections applied to 
the observed altitude.* These we shall now enumerate and 
fsxplain. 



* The observed altitude is obtained by means of an instrument called a quad- 
rant of reflection, or simply a quadrant. This instrument is a frame of wooc| 
in the form of a sector of ^ circle^ the arp of which is graduated to degrees find 
parts of a degree. This frame is suspended so that the plane of the circle shall 
be TerticiJ. It has an arm, one extremity of which is attached to the centre of 
^he circle, and which is moveable about this point; upon this arm is a small 
mirror, and opposite to it is a plane glass half of which is mirror and half trans? 
parent. When a heavenly body, seen by double reflection in these two mirrors, 
is brought by the movement of the arm, upon which one of the mirrors is placed, 
to coincide with the line of the horizon at sea as seen through the trapsparent 
p«rt of the opposite glassi, the outer extremity of the arm points out upon the 
graduated arc the number of degrees of altitude of the heavenly body above tho 
horizon. 

The construction of this instrument depends upon the optical principle that 
the angle of incidence is equal to the angle of reflection. The angular move-i 
ment of the image of the heavenly body is double the angular movement of the 
arm so that to measure the greatest altitudes the limit of which is 90°, the gradu- 
ated arc needbe butthe eighth of a circumference j thedegrees upon itare however 
numbered as if it were a quadrant to save the trouble of doubling them. The in- 
ftrument takes its name from theaippuut which it measures, instead of from th^ 
magnitude of its arc. There are colored glasses attached, which can be inter- 
posed so that the rays of light, coming from the heavenly body to the eye, can 
ht made to paia through them when taking the altitude of the sun. 



NAUTICAL ASTRONOMY. 



153 




Of the Dip or the Depression of the Horizon, 

(106.) Let E represent 
the place of the observer's 
eye, and s the situation 
of any celestial body; the 
first object is to obtain 
its apparent altitude 
above the horizontal line 
EH ; that is, the angular 
distance seh. Now. as 
to the observer, the visi- 
ble horizon is ebh', the 
altitude given by the 
instrument is the angle s£;h' ; hence we must subtract from 
this observed altitude the angle heh', called the Dip or De^ 
pression of the Horizon^ in order to obtain the apparent alti- 
tude SEH. 

The angle heh', or its equal c, is calculated for various 
elevations, 4e of the eye above the surface of the sea ftom the 
proportion, 

ce : EB == Vec^ — CB^ : : rad. : sin. c ; 
find the results are registered in a table, (Table IX-) 

Of the Semidiameter. 

(107.) When the foregoing correction for dip has been 
applied, the result will be the apparent altitude of the point 
observed above the horizontal plane through the observer's 
eye. If this point be the uppermost or lowermost point of the 
disc of the sun or moon, a further correction will be necessary 
to obtain the apparent altitude of the centre ; that is, we must 
apply the angular distance due to the semidiameter ; by which 
we mean, the angle at the eye of the observer subtended by 
the semidiameter or radius of the sun or moon. This quan- 
tity, which is continually varying, both for the sun and moon, 
in consequence of the variation of the distance, is given in 



154 APPLICATION OF TRiaONOMETRY. 

the Nautical Almanac for every day in the year. But in the 
case of the moon the semidiameter itself requires a small 
correction depending upon the observed altitude. For the 
semidiameter, furnished by the Nautical Almanac, is the appa- 
rent horizontal semidiameter, or the angle, it subtends when 
in the horizon ; but as the moon approaches the zenith, her 
distance from the observer diminishes, and therefore her 
semidiameter is viewed under a greater angle. As she is 
nearer to the observer when in the zenith than when in the 
horizon, by one semidiameter of the earth, and as her dis- 
tance from the earth's centre is about 60 semi-diameters of 
the earth, the horizontal semidiameter will in the zenith be- 
come increased by about W part, and at intermediate eleva- 
tions the increase will be as the sine of the altitude. On this 
principle is formed the Table at the end, entitled Augmenta^ 
tion of the MoovUs Semidiameter^ (Table XII.) and contain- 
ing the proper correction to be added to the given horizontal 
semidiameter to obtain the true semidiameter. 

On account of the great distance of the sun, no such cor- 
rection pf his semidiameter is necessary. 

The corrections for dip and semidiameter being thus ap^ 
plied, the result is called the apparent altitude of the centre, 
■In the case of the stars the only correction for the apparent 
altitude is the dip. It must however, be here remarked, that 
if the centre of the object were visible, and its altitude, instead 
of that of the limb, were to be taken, we should not, after 
applying the correction for dip, obtain precisely the same 
result as that which we have just called the apparent altitude 
of the centre, but should get a value somewhat less. The 
reason of this is, that every vertical arc iji the heavens is 
shortened by refraction^ as we shall shortly explain, so that 
the centre would not exceed the observed altitude of the lower 
limb, or fall short of that of the upper, by so great a quantity 
as the true semidiameter. Hence, from the apparent altitude 
"^f the centre, as found from applying the true semidiameter to 
le apparent altitude of the limb, a small quantity should in 
rictness be subtracted, and this small correction becomes 



fiAUTlCAL ASTRONOMY. 155 

liecessary when the longitude is to be determined witli accu- 
racy. This correction was first proposed by Dr. Thomas 
Young. A table for it is given at the end. (Table XI.) 

To obtain the true altitude requires two other corrections^ 
viz. for refraction and for parallax. The former of these 
has indeed an effect upon the two preceding corrections, dip 
6.nd semidiameter, which require certain modifications in 
consequence. One of these we have adverted to above, and 
the other will be noticed more particularly in the following 
article. 

Of Refraction. 

108. It is a universal fact in optics, that if a ray of light 
pass obliquely out of one medium into another of greater 
density, it will be bent out of its original direction at the 
point where it enters the new medium, and proceed through 
it in a direction more nearly perpendicular to its surface at 
that point. Hence the rays of light, proceeding from the ce- 
lestial bodies, become bent downwards as soon as they enter 
the atmosphere, their course beingdirected more nearly towards 
the centre of the earth, so that the rays which enter the eye 
of an observer, ^nd by which any celestial object becomes 
visible to him, would, if not thus bent down, pass* over 
his head ; the object is therefore seen by him above its true 
place ; the angle between this apparent direction and the true 
direction of the object, measures the refraction ; and, like the 
correction for dip, it is always subtractive; it increases from the 
horizon, where it is greatest, to the zenith, where it vanishes^ 
as the rays from objects iir the zenith enter the atmosphere 
perpendicularly. 

It is the refraaiion which causes the sun and moon, when 
near the horizon, to present sometimes an elliptical appear- 
ance, the vertical diameter (and, indeed, every oblique dia- 
meter) seeming to be shorter than the horizontal, because the 
lower limb, or edge, being more elevated by refraction than 
the upper, the two are brought, in appearance, more nearly 
together. 

At the end of the volume we have given a table of refrac* 



186 iPPLICAtlON OP TRlOONOMKTftir^ 

tions for different altitudes, from the horizon to the zenitH,- 
and adapted to the mean state of the atmosphere, (Table VIII.) J 
but, as the actual state of the atmosphere generally differs from 
this, it becomes necessary, where the true altitude of the body 
is i^uired with the Utmdst acchracy, 16 apply a correction td 
the numbers in this table, so as to adapt them to the existing 
temperature and density of the atmosphere at the time of obser- 
vation, as indicated by the thermometer and barometer. This 
table of corrections is annexed to the table of mean refrac- 
tions. It should, however, be observed that below 4P the re- 
fraction is very variable and uncertain, and such low alti- 
tudes should be avoided as much as possible at sea. 

It will be unnecessary to use this annexed table for cor^ 
recting the altitude of a celestial object when the latitude of* 
the ship is the only object of the observation, as such a cor- 
rection could seldom make a difference so great as half a mile 
in the resulting latitude ; but, in determining the longitude 
by the Lunar Observations^ the neglect of these small cor- 
rections would sometimes introduce an error in the resulting 
longitude of more than thirty miles. 

It should be remarked here, that the dip, as determined in 
article (106), is on the supposition that refraction has hot ele- 
vated the apparent horizon, but as such is not the case, the 
dip requires a correction ; the amount of this correction is 
very uncertain, on account of the irregularity of the hori- 
zontal refractions, although it is unquestionable that some 
correction is requisite. It is usual to allow about i or tV of* 
, the computed dip for the correction. In our table tV is al- 
lowed, which is according to Dr. Maskelyne, but liambert 
and Legendre make it tV* ^ 

When the foregoing corrections have been applied to the 
observed altitude, the result will be the true altitude of the 
centre above the visible horizon, and it now remains to apply 
the correction necessary to reduce this to the true altitude of 
the centre above the rational horizon ; that is, to the altitude 
'^hich the body would have if the observer were situated at 

9 centre of the earth instead of on its surface. 




kAVriclL A^TROKOMf; 117 

Of the Parallax. 

I09i In order to explain the 
hature and effect of parallax^ 
let s represent the place of the 
object 'observed frcJm the sur- 
face of the earth, at e ; then 
the angle seh, that is, the ob- 
served angle when corrected 
for dip semidiameter, and re- 
fraction, will be the true alti- 
tude of the object, in reference 
to the observer's sensible horizon eh ; and the angle sch will 
be the true altitude, in reference to the rational horizon ch j 
and the difference of these angles is the parallax in altitude* 
If the body be at h, in the sensible horizon, then the differ- 
ence of which we speak is the entire angle hcb ; this is call- 
ed the horizontal parallax. 

Since the angle se'h is equal to the angle scr, we have foi 
the parallax in alt., se'h — seh = esc ; that is, the parallax 
is the angle which the semidiameter of the earth subtends at 
the object ; it is obviously greatest in the horizon, and nothing 
in the zenith, and is the quantity which must be added to the 
true altitude above the sensible horizon to obtain the true 
altitude above the rational horizon. 

The sun's parallax in altitude is given in a Table at the end j 
(Table X.) and the moon's horizontal parallax is given for the 
noon and midnight at Greenwich, of every day of the year, iii 
the Nautical Almanac ; and from the horizontal parallax thus 
obtained the parallax iii altitude must be calculated. This 
is easy \ for since in the triangle sec^ we have the propor- 
tion 

so : EC : : sin sEC=sin sez=cos seh : sin iE:sc ; 

it foUoTH^s that the sine of the parallax in altitude varies ad 
the cosine of the altitude ; but when the altitude = as if 
the case of horizontal parallax cos. altitude =? rad. hence i 
tad. is to the cosine of the altitude^ so is the sine of the h<M 

S4 



158 APPLICATIOir OF TRIOONOJUETIlTy 

zontal parallax, to the sine of the parallax in altitude. In 
other words, the log. sine of the horizontal parallax, added 
to the log. cosine of the altitude, abating 10 from the index^ 
will give the log. sine of the parallax in altitude ; but as the 
parallax is always a very small angle, it is usual to substitute 
the arc for its sine, so that 

log* hor. par. in seconds + log. cos. alt. — 10 = log. par. in alt^ 

in seconds^ 

We must observe here that the horizontal parallax, given 
in the Nautieal Almanac, is calculated to the equatorial 
radius of the earth } and, therefore, except at the equator, a 
small subtractive correction will be necessary, on account of 
the spheroidal figure of the earths A table of such correc 
tkms is given at the end. (See Table XIII.) 

110. Such arc the corrections necessary to be appKed to» 
the observed altitudes of celestial objects, in order to obtain 
their true altitudes. A few other preliminary, but very sim- 
ple, and obvious operations mast also be performed upon the 
several quantities taken out of the Nautical Almanac, ia 
order to reduce them to their proper value at the lime and 
place of observsation ; for the elements furnished by the 
Nautical Almanac are computed for certain stated epochs, 
and their values for any intermediate epoch must be found 
by proportion. But ample directions for these preparatory 
operations are contained in the " Explanation of the Articles 
in the Nautical Almanac," by the late Dr. Maskelyne,, which 
accompanies every edition of that work. 

Examples of the Corrections. 

!• On the 14th of January, 1833, suppose the observed 
altitude of the sun's lower limb* to be 16° 36' 4", the €*server's 
eye to be 18 feet above the level of the sea, the barometer to 
stand at 29 inches, and the thermometer at 68° : required the 
true altitude of the sun's centre. 



mim^fmmm'm00m 



• Tiie limb of tbe suii or raoon is the eircumfeMned of Um diM. 



WAtJTICAL ASTROirOMY. . 159 

Observed alt ^s L, L, 
Depression of the horizon 

App. alt. of L. L« « « • < 
Refraction ««...< 
Correction for Barometer 
Correction for Thermometer , 

True alL of L, K above the visiUe horizon 16 28 36.3 
Sun's semidiameter (NauL Aim.) « « -f- 16 175 
Parallax in altitude ... - -f- 8.4 



16036' A" 


4 4 


16 32 


- 3 14 


— 6.6 


3.2 



True altitude of sun's centre . . . 16 45 2. 

2. On the 20th of May, 1833, suppose that in longitude 
5ibout 77° 30' west, and laL about 48° north, at 3* a{4)ii3:ent 
time, the altitude of the moon's lower limb is observed to be 
18° 8' 34", the height of the eye being 20 feet, the barometer 
28.5 inches, and the thermometer 46° .: reqwirad the irue alti- 
tude of the sun's centre. 

Here the object being the moon, it will be necessary to com- 
pute the parallax in altitude, from having the horizont&i 
parallax corresponding to the time at Greenwich. 

The horizontal parallax is given in the Nautical Ainmnac 
for every noon and midnight ; and, therefore, to find it for 
any other intermediate time, we must say as 12* is/toite 
variation in 12*, so is the proposed time to the variation 4lue 
to that time. 

In like manner must the moon's semidiameter be reduced 
by proportion to the time of observation, since it sensibly 
varies in the course of a few hours. We shall begin, there- 
fore, with finding in this way the true horizontal mraJlai: 
and semidiameter for the time of observation, reduced to the 
meridian of Greenwich. 

Longitude of the ship in time 5* 10* after Greenwich tinMj. 
Apparent time at ship 3 

Apparent time at Greenwich 8 10 



160 



APPLICATION QF TR|GONOBfETRT« 



Hor. par. at noon (Naut Aim.) 68' 17 
Hor. par. at midnight ... 58 31 



Variation in 12a . . . . 


. 14 






/. m .- 8* 10« : : 14" : . . 
Hor. par. at noon . . 


9.5'^ 
58' 17 


^or. par. at reduced time . 


. 58 26.5 
60 


pitto in seconds .... 
pirn, of par. for lat. 48^ . 


3506.5 
— 6.3 


^rae hor. parallax . . . 


3500.2 



Semidiam. at noon 
Semidiam. at midnight 



15 57 



Variation in 12A 4 



.-. 12* : 8* 10» : j 4" j . 
Semidiameier at noon . 



2.7'' 
. 15' 63 



Hor. semidia. at reduced time 15 65.7" 
Augmentation for 18^ alt. . 6.2 

True semidiameter ... 16 0.9 



For the Apparent Altitude. 



Observed altitude of D's L. L. . . 

Depression 

Semidiameter minus contraction* . 

Apparent alt. i)'s centre 



r • • 1 



18° 8' 34" 
— 4 17 
16 67.9 

18 20 14.9 



For the Parallax in Altitude, 



pos. f)'s app. alt. 
hor, parallax 

Par. in altitude 



180 20' 15" 
3500.2 " log. 



9.977367 
3.544093 



3322.6" .... 3.52 1 46U 



For the true Altitude. 

Apparent alt. of c's centre . . 

Refraction 

Barometer ....,.,, 
Thermometer ,..,... 

True alt. above sensible horizon 
Parallax in altityde 3322.5" : 
True alt. of ©'s pentre .... 



18° 20' 14.9'' 
— 2 54.2 
— 8,8 
4- 1.4 

18 17 13.3 
: + 65 22.6 



, . • 19 12 35.8 
These two examples will serve for specimens of the coTt 
yeptions to be applie(i to an observed altitude, in order to 
deduce from it the true altitude of the body's centre. In the 
case of the moon, the corrections, when the utmost accuracy 
}s sought, are rather numerous, as the last example shows, 
' 

• TH^ contraction is obtained fpm T«bla XI. 



NAUTICAL ASTRONOMY. 161 

Bat, in finding the latitude at sea, it is usual to dispense with 
some of these, more especially with the corrections for tem^ 
perature, for the contraction of the moon's semidiameter, and 
for the spheroidal figure of the earth ; because an error of a 
few seconds in the true altitude will introduce no error worth 
noticing in the resulting latitude. When, however, the 
object of the observers is to deduce the longitude of the ship, 
all the data, furnished by observation, should be as accurate 
as possible ; for the problem is one of such delicacy that by 
neglecting to allow for the influence of temperature would 
alone introduce in some cases an error of from 30 to 40 miles 
in the longitude. 

When the object observed is a star, several of the forgoing 
corrections vanish : the only corrections, in this case requi^ 
site, are those for dip and refraction, modified as usual for the 
temperature. 

Ill, To determine tlie latitude at sea from the meridian 
altitude of any celestial object whose declination is known. 

The determination of the latitude, by a meridian altitude, 
js the easiest, and in general the safest, method of finding the 
ship's place on the meridian ; for both the observations and 
the subsequent calculations being few, they are readily per- 
formed, and with but little liability to error in the result ; this 
method, therefore, is always to be used at sea, unless foggy or 
cloudy weather render it impracticable. 

The declination of the object observed is supposed to be 
given in the Nautical Almanack for the meridian of Green-^ 
wich ; it may therefore be reduced to the meridian of the ship^ 
by means of the longitude by account, which will always be 
sufficiently accurate for this purpose, although it should difler 
very considerably from the true longitude, because declination 
changes so slowly that even an error of an hour in the longi*^ 
tude would cause an error in the declination too s^i$tll to 
deserve notice. 

Having then thus found the distance of the object from the 
equinocti^, and having, by means of the observed altitude 
properly corrected, obtained the distfoice of the same objec 



162 



APPLICATION OF TRIGONOMETRY. 



from the ship's zenith, the distance of the zenith from the 
equinoctial, that is, the latitude, immediately becomes known. 

1. Let s be the object observed, the zenith z being to the 
north of it, and the object itself north of the equinoctial eq, 
then the latitude ez is equal to the zenith distance, or coalti- 
tude zs + the declination, and it is north. 

2. Let s' be the object, still north of the equinoctial, but so 
posited that the zenith is south of it, then the latitude ez is 
equal to the difference between the zenith distance s'z, and 
declination s'e, and is still north. 

3. Let now the object be 
fit s", south of the equinoctial, 
and the zenith to the north of 
the object, then the latitude 
BZ is equal to the difference 
between the zenith distance 
«"z and declination s"e, and 
it is north. 

We have here assumed the 
north to be the elevated pole, 
but if the south be the eleva- 
ted pole, then we must write south for north, and north for 
south. Hence the following rule for all cases. 

Call the zenith distance north or south, according as the 
zenith is north or south of the object. 

If the zenith distance and declination be of the same name, 
that is, both north or both south, their sum will be the lati- 
tude ; but, if of different names, their difference will be the 
latitude, of the same name as the greater. 

examples. 
I. If on tlie 2d of May, X833, the meridian altitude of the 
sun's lower limb be 47° 18', height of the eye 20 feet, and 
iongitude by account 32° E. : required the latitude, the sun 
J[)eing south at the time of observation. 

Observed alt. of ©'s L. L. . 47° 18' 0" 
Dip. of the horizon . . — 4 17 




App. alt. of &s L. L. 



46 13 43. 



KAtTlCAL ASTROlfOMtr. 16$ 

The longitude in time is 2^ 8* east, so that the time at 

Greenwich is 2* 8* before the noon of the 2d of May ; hence, 

to find the corresponding declination, we hate by the Nautical 

Almanac^ 

24*:2*8«:?18 r:r38"; 

so that, 1' 38', the variation in 2* 8", must be subtracted from 
15° 23' 21" N. the declination of the sun on May 2, at noon } 
hence the proper declination is 15° 21' 43" N. 

Observed alt. of m^s L. L. . 47° 18' 0" 
Dip — 4 ir 

App. alt. of ©'s L. L. ... 46 13 43 

Refraction - — 56 

Parallax ....... +6 

Semidiam. (Naut. Aim.) . . 15 53 

True alt. of ©'s centre . . 46 28 46 

Zenith distance 43 31 14 N- 

O's declination 15 21 43 N. 



•*«- 



Latitude 58 52 57 N. 

2. On the first of January, 1820, the meridian altitude of 
Capella was 27° 35', the zenith being south of the star, and 
the height of the eye 22 feet ; required the latitude. 

Observed altitude . . 27° 35' 0' 
Dip — 4 30 

Apparent altitude . . 27 30 30 
Refraction . . - . . — 1 51 

True altitude .... 27 28 39 
Zenith distance ... 62 31 218. 
Star's dec. (Naut. Aim.) 45 48 39 N. 

Latitude 16 42 42 S. 

3. On the 19th of February^ 1833, the ship being in longi« 
lude ifp W., the observed meridian altitude of the moon's 



161 APPLICATiON OF TRIGOICOMETHYi 

lover limb was 65^ 6' ; the zenith north of the moon : cLnd 
the height of the eye 16 feet : required the latitude. 

Here the time of observation at the ship is not given, it 
must therefore be calculated, and we have these data for this 
purpose, viz., that the ship is 40° W. of Greenwich, and that 
the moon is on its meridian. The following process there- 
fore immediately suggests itself. 

The moon passed the merid. of Greenwich Feb.l9(Naut.Alm() 

6* 56"« 0' 
. Feb.20 » 7 59 



Interval between the two passages » . , 24 -f I 3 0. 

Hence I* 3"* is the moon's retardation in 25* 3*», and, by pro- 
portion using for the longitude 40°W., its value in time 2* 40*», 
we have, 

25* 3~: 1*3"«: : 2*40«: 0^6~42'; 
that is, the moon is retarded 6"* 42' in passing from the me-^ 
ridian of Greenwich to that of the ship, and therefore, in- 
stead of the apparent time at the ship being 6* 56**, as it ne- 
cessarily would be if there were no retardation, it will be 6* 
42' later. Hence 

Apparent time at the ship » 7* 2'»42* 
Ship's longitude W. . » . 2 40 

Time at Greenwich .» * 9 42 42 



Having ^hus got the apparent time at Greenwich when thd 
observation was made, we may, by a reference to the Nauti' 
cat Almanac and a subsequent proportion, find the moon's 
declination at that time : thus 

Mooifs declination at Greenwich,Feb.l9 at noon 26° 38' 17" 

Feb. 19 at midn. 26 54 39 



Change of declination in 12 hours . . 16 22« 

... 12* : 9* 42~42' : : 16' 22" : 13' 15 " ; 
hence 13' 15' is the amount of the change of declination, 
from noon to 9* 43"«, on the supposition, however, that the 
motion of the moon in declination may be considered as 



kAtTiCAL AS^RONOSlt; l65 

fe^iiable during the twelve hours. But on account of the ir- 
regular motion of the moon, this supposition introduces a 
sensible error, which may however be corrected by means of 
the table of " Equation of Second Differences," given in the 
Nautical Almanac, and explained by Dr. Maskelyne's accom- 
panying " Explanation." The correct change of decliuatioii 
is thus found to be 14' 16'. But from the year 1833, the de- 
clination of the mooii will be given in the Nautical Almanac 
to every three hours, and the change for any shorter interval 
may then be obtained with the requisite accuracy by propor- 
tion, as above. Taking in the present case 14' 16" for thd 
correct changCj we have 

Declination for preceding nbOil ■. ^6^28' 17" N. 

Increase of Declination . . 14 16 



Declin. at the time of observation ; 26 52 33 N; 



iBefbre we can 6nd the proper correction for parallax, wd 
must deduce the apparent altitude of the centre. 

Observed altitude of '©s L. L. . 55° 6' 0" 
Dip. . . . i ; ^ _ 3 50 

Semidiameter (Naut. Almi) . i 16 13 

Augmentation of semidiam.for 55°of alt. 13 

Apparent alt. of i)'s centre . 66 18 36 cos. 9.756217 

Hor.par.in seconds at 9*43«» (Naut. Aim.) 3672 log. 3.562911 

Parallax in altitude in seconds . 2033 log. 3.308128 

therefore the correction for parallax is 33' 63"i 

Having thus reduced all the corrections to the time of ob^ 

servation, we readily obtain the true altitude, and thence the 

latitude as follows : 

Apparent alt* of i)'s centre * 65° 18' 36" 
Refraction . . . — 40 

Parallax in altitude . * 33 63 



True altitude i ; 55 61 49 



35 



166 APPLICATION OF TRIGONOMETRY^ 

Zenith distance # . ^ 34 8 IIN^ 
Declination . . . 26 52 33 N. 



liatitude . . . . 61 44 N. 



SCHOLIUM. 

Theae examples will, no doubt, be found sufficient to put 
the student in possession of the method of applying the va^ 
lious corrections to the observed meridian altitude of a celes^ 
tial object, in order to deduce from it the latitude of the ship^ 
Biit it should be remarked, that in most works on Nautical 
Astronomy, subsidiary tables are inserted for the purpose of 
abridging some of the foregoing corrective operations ; such 
tables, therefore, oflFer very acceptable aid to the practical 
navi^tor. The most esteemed works of this kind are Dr. 
Maekay's "Treatise on the Theory and Practice of finding 
the Longitude at Sea'^; the "Nautical Tables" of J. De Men- 
doza Rios, Mr. Riddle's book on Navigation and Nautical 
Astronomy, and the work of Dr. Bowditch. 

It should also be observed here, that in the preceding ex-^ 
amples the celestial object is supposed to be on the meridian 
abtyve the pole ; that is, to be higher than the elevated pole. 
But, if a meridian altitude be taken below the pole, which 
may be done if the object is circumpolar, or so near to the 
elevated pole as to perform its apparent daily revolution about 
it without passing below the horizon, then the latitude of the 
place will be equal to the sum of the true altitude, and the 
codeclination or polar distance of the object ; for this sum 
will obviously measure the elevation of the pole above the 
horizon, which is equal to the latitude. 

112. To determine the latitude at sea, by means of two 
altitudes of the sun, and the time between the observations* 

In the preceding article we have shown how to determine 
the latitude of the ship by the meridian altitude of the sun, or 
of any other heavenly body, whose declination may be found. 
But, as already remarked, the object we wish to observe may 



RATTTICAL ABTRONOMY. 167 

be obscnred when it comes to the meridian, and this may 
happen for many days together, although it may be frequently 
visible at other times of the day. As therefore the opportu- 
nity for a meridian observation cannot be depended upon, it 
becomes an important problem to determine the latitude aC 
eea, by observations made out of the meridian ; and conside- 
rable attention has accordingly been paid, by scientific per- 
sons, to the method of finding the latitude by " double alti- 
tudes," and various tables have been computed to facilitate 
the operation. But the direct method, by spherical trigono- 
metry, though rather long, involving three spherical triangles, 
will be more readily remembered, and more easily applied by 
persons familiar with the rules and formulas of Trigonometry, 
than any indirect or approximate process ; we shall therefore 
explain the direct method. 

Let p be the elevated pole, 
z the zenith of the ship, 
and s,s', the two places of the 
sun, when the altitudes are 
taken. Then, drawing the 
great circle arcs as in the 
figure, we shall have these 
given quantities, viz. the co- 
declinations PS, ps' ; the co- 
altitudes zs, zs', and the hour 
angle sps', which measures 
the interval between the observations; and the quantity 
Bought is the colatitude zp. Now, in the triangle pss', we 
have given two sides and the included angle to find the third 
side ss", and one of the remaining angles, say the angle pss'^ 
In the triangle zss' we have given the three sides to find the 
angle s'sz ; having then the angles pss', s'sz, the angle zsp 
equal to their difference, becomes known, so that we have, 
lastly, two sides and the included angle in the triangle zsp, to 
find the third side zp. 

Before the application of the trigonometrical process, the 
observed altitudes must, of course, be reduced to the tru# 




J.68 APPLICATION OF TRIGONOMETRT. 

altitudes, as in the preceding examples. Moreover, as the ship 
most probably sails during the interval of the observation, an 
additional reduction becomes necessary; the first altitude 
must be reduced to what it would have been if taken at the 
place where the second was taken 2 this correction will be 
known if we know the number of minutes or miles which 
the ship has made, directly towards or directly from the sun, 
since leaving the place where the first observation was made. 
To find this, take the angle included between the ship's 
course and the sun's bearing, at the first observation ; and 
considering this angle as a course, and the distance sailed as 
the corresponding distance, find by the traverse table, or by 
tile operation of plane sailing, the difference of latitude, which 
will be the amount of the approach to, or departure from, the 
sun. This must be added to the first altitude if the angle is 
less than 90°, because the ship will have approached towards 
the sun ; but it must be subtracted when the angle exceeds 
9(P. If the angle is 90°, no correction for the ship's change 
of place will be necessary, since in that case sailing in a per-r 
pendicular to the direction of the sun, she maintains always 
the same distance from him, 

Where great accuracy is aimed at, account should be taken 
of the ship's change of longitude during the interval of the 
observations ; when converted into time it must be added to 
the interval of time between the observations when the ship 
has sailed eastward, and subtracted when she has sailed 
westward. This correction is very easily applied. 

» Having thus mentioned the necessary preparative correc-? 
tions, we shall now give an example of the trigonometrical 
operation. 

EXAMPLES. 

Let the two zenith distances corrected be (see leist fig.) zs = 
rso 64' 13", zs' = 470 45' 51", the corresponding declinations 
8^ 18' and 8° 15' north, and the interval of time three hours j 
to determine the latitude, 



NAUTICAL AS'TRONOMT. 160 

Considering ss' to be the base of an isosceles spherical tri* 
angle, of which one of the equal sides is J (ps + ps')*= 81° 
43' 30", and the vertical angle equal to 3* or 45°, let the per* 
pendicular pm be drawn, then we have in the triangle pmb 

right angled at m, ps=81o 43' 30", and p = ^ = 2299^] 
given to find sm = ^ ss' as foUowp. 

1. To find SB' from the triangle pms, 

sin pst 81043' 30'' 9.996455 

sin p 22 30 9.582840 



sin SM 22 15 11.4 9.578295 



ss'= 44 30 22.8. 



II. To find V8s' from the triangle pss', 

sin. ss' 440 30' 22.8" , arith. comp. 0.154290 
3in.PS' 81 45 . . . . , . 9,995482 
sin. sps' 45 9.849486 



sin. pss' 86 38 63 9.999257 



This angle is acute like its opposite side, (see Art. 128.) 



* Which we may, without sensible error, where the base is so smalL 
t The proportion employed here is that of Art 38 ; b is understood as the first 
^rm of the proportion. 



170 APPLICATION OF TRIGONOMETRY. 



III. To find zss' in the triangle zss'. 

zs' 470 45' 51" 
sin^ zs 73 54 13 . . arith. comp. 0.017369 
sin. ss' 44 30 22.8 . arith. comp. 0.154290 

2) 166 10 26.8 



i sum = 83 5 13.4 

sin. (i sum— zs) 9 11 0.4 9.203022 

8in.(Jsum— ss') 38 34 50.6 9.794918 

2) 19.169599 

sin. I zss' 22 36 26.4 9.584799 

.-. zss' = 450 12' 52.8" 
pss'=:86 38 53 



psz = 41 26 0.2 



IV. To find the two unknown angles of the triangle zsp. 

COS. i (zs+ps) 77o48'52" ar. comp. 0.675555 ar. comp. sin. 0.009897 

•co8.Hm»^p«) 7o49'17'' 9.995941 sin 9.133811 

cot. I PSZ 20O43' 3" . . . . 10.422276 10.422276 

t9Xk,i{zPM'^vzB) 85o23'35" .... 11.093772 tan J(pt-«)20ol2'32"3.565984 

20oi2'32" 



8ZP = 105^36' 7" 
zps = 65011' 3" 



V. To find zp in the triangle zsp. 



COS. i ( z— p) 20° 12' 32" ar. comp. 0.027593 

COS. I (z + p) 85° 23' 31" . . . 8.904822 

tan. i (zs+ps) 770 48' 52" . . . 10.665658 

tan. i zp 210 37' 14" 9.598073 
zp = 430 14' 28" 



* This sign -^^ is employed to express the difTerence between two quantities 
whiefaever may be the greater. 



NAI^TICAL ASTRONOMY* 171 

Upon the same principles may the latitude be determined 
from the altitudes of two fixed stars, taken at the same time ; 
in this case s, s', in the preceding figure, will represent the 
two stars : ps, ps', their known polar distances, and the angle 
BPs', the difference of their right ascensions ; the same quan* 
titles are therefore given as in the case of the sun, but, as in 
the case of two stars ps, ps', may difier very considerably ; 
8s' cannot be considered as the base of an isosceles trianglei 
but must be computed from the other two sides and their in- 
cluded angle. In the Nautical Almanac for 1825, Dr. Brink- 
ley has computed for 1822, and tabulated, the distances SS' 
for certain pairs of stars, conveniently situated for observa- 
tion, and has annexed the change of distance corresponding 
to 10 years. The same table shows also the difference 
of right ascension for each pair of stars, with the change in 
10 years ; so that by help of this table the computation for 
finding the latitude from the simultaneous altitudes of two 
fixed stars becomes considerably abridged. 

For other methods of determining the latitude, the student 
may consult "Mackay on the Longitude," Vol. 1., and Cap- 
tain Kater's Nautical Astronomy, in the Ency. Metropoli- 
tana, &c. 

On finding the Longitude by the Lunar Observations. 

113. There are several astronomical methods of deter- 
mining the longitude of a place which cannot be accurately 
employed at sea, on account of the great difiiculty of mana- 
ging a telescope on shipboard ; we shall not, therefore, enter 
here into any explanation of these methods, but shall confine 
ourselves to the lunar method of determining the longitude, 
which is justly regarded as the principal problem in Nautical 
Astronomy. Before entering upon the solution of this pro- 
blem, it will be necessary to make a few introductory remarks. 

The determination of the longitude of a place always re- 
quires the solution of these to problems, viz. : 1st, to deter- 
mine the time at the place at any instant ; and, 2d, to deter- 
mine the time at the first meridian, or that from which th^ 



172 APPLICATION OF TRIGONOMEtRlT. 

longitude is estimated^ at the same instant ; for the difference 
of the times converted into degrees, at the rate of 15^ to an 
hour) will obviously give the longitude. 

When the latitude of the place is known, (and it may be 
found by the methods already explained,) the time may be 
computed from the altitude of any celestial object whose de- 
clination is known ; for the coaltitude, codeclination, and co^ 
latitude) will be three sides of a spherical triangle given to 
find the hour angle, comprised between the codeclination and 
the colatitude. But to find the time at Greenwich requires 
the aid of additional data, besides |;hose furnished by observa- 
tions made at the place. The Greenwich time may, indeed, 
be obtained at once, independently of any observations at the 
place, by means of a chronometer, carefully regulated to 
Greenwich time, provided it be subject to no irregularities 
after having been once properly adjusted. A ship furnished 
with such a timepiece always carries the Greenwich time 
with her*, and the longitude then becomes reduced to the pro- 
blem of finding the time at the place. Chronometers are 
now brought to such a state of perfection that very great de- 
pendence can be placed on them, and they are accordingly 
always taken out on long voyages for the purpose of show- 
ing the Greenwich time, and are thus of great use to the 
mariner. Still, however, as the most perfect contrivance of 
human art is subject to accident, and the more delicate the 
machine tiie more liable is it to disarrangement, from causes 
which we may not be able to control, it becomes highly de^ 
sirable, in so important a matter as finding the place of a ship 
at sea, to be possessed of methods altogether beyond the in-' 
fluence of terrestrial vicissitudes, and such methods the celes- 
tial motions alone can supply. 

The angular motion of the moon in her orbit is more rapid 
than that of any other celestial body, and sufficiently great 
to render the portion of its path passed over in so short a 



• As chronometers show mean time, the equation of lime must be applied to 



btain the appartrU time at Greenwich. 



NAUTICAL ASTEONOMY. 173 

time as two or three seconds, a measurable quantity even 
with a small portable instrument (the sextant.)* 

It is obvious, therefore, that if the distance of the moon's 
centre from any celestial body, in or near her path, be com- 
puted for any Greenwich time, and this distance be found the 
same as that given by actual observation at any place, then 
the difference between the time of observing the phenomenon 
and the time at Greenwich, when it was predicted to happen, 
will give the longitude of the place of observation. Now in 
the Nautical Almanac the distances of the moon from the 
sun, and from several of the fixed stars near her path, are 
given for every three hours of apparent Greenwich time, and 
for several years to come ; and the Greenwich time, corres- 
ponding to any intermediate distance, is obtainable by simple 
proportion with all requisite accuracy ; so that by means of 
the Nautical Almanac we may always determine the time at 
Greenwic]^ when any distance observed at sea was taken. 

The distances inserted in the Nautical Almanac are the true 
angular distances between the centres of the bodies, the observer 
being considered as at the centre of the earth, and to the true 
distance therefore every observed distance must be reduced ; 
it is this reduction which constitutes the trigonometrical diffi- 
culties of the problem ; and it consists in clearing the lunar 
distance Jrom the effects of parallax and refraction ; how to 
do this it is now our business to explain. 



* The seztant is constructed upon the same principles as the quadrant It is 
usually of brass, is made to hold in the hand, whereas the quadrant is suspended 
at the centre. It measures 120°, having an arc equal to -)- the circumference, from 
which unlike the quadrant it takes its name. The angular distance of two hea- 
Tenly bodies apart is obtained by making the reflected image of one coincide 
with the other as seen directly. 



26 



lU 



APPLICATION OF TBIOONOMETRT. 




Let m, Sf be the ob- 
served places of the 
moon and sun, or of the 
moon and a fixed star, 
and let m, s, be their true 
places. M will be above 
f7ii because the moon is 
depressed by parallax 
more than it is elevated 
by refraction ; but s will 
be below s^ because the 
sun is more elevated by 
refraction than it is depressed by parallax. Observation 
gives the apparent distance ms, and the apparent zenith dis- 
tances zm, zs : by appljring the proper corrections to these 
latter we also deduce the true zenith distances zm, zs, and 
with these data we are to determine the true distance, ms, by 
computation. 

Put d for the apparent distance.* 

D true distance. 

a, a apparent altitudes.* 

A, a' true altitudes. 

Then in the triangle mzs, we have, (Art. 82,) r being 1, 

cos D — sin A sin a' ^ 

cos z = ■■ 7 :t 

cos A cos A ' 

and in the triangle mzs, 

cos d — ^sin a sin a' 

cos z = -. : 

cos a cos a 

hence, for the determination of d, we have this equation, viz., 
cos D — sin A sin a' cos d — sin a sin a! 



cos A COS A 



cos a cos a' 



* In observing d with the sextant, it is the nearest point of the limb of the 
moon which is made to coincide with the other heavenly body, and in obserriug 
a with the quadrant, it is the limb also which is made to coincide with the hori- 
zon; 80 that d and a must be corrected for the semidiaterof the moon; similar 
remarks apply to the sun, if he be the other heavenly body. 

t Observe that ▲ and a' are the complements of zm and zs. 



N 



HAUTICAL ASTRONOMY. 176 

from which we immediately get 

(« . • i\ cos A COS A . « • , 

COS a — sm a sm a) — — r+si^ a sm a' 
' cos a cos a 

But cos a cos a' — sin a sin a' = cos (o+a') Art. 69 ; 

transposing cos a cos a' and substituting the value of — sin a 
sin a' thus obtained, we have 

cosrf+cos(a+a') — cos a cos a' , , . . , 

cos D = ■ ^ — ^---^ — ; cos ACOSA'+Sm AsmA' 

cos a cos a 

Dividing the last term of the numerator by the denomina- 
tor, the quotient is — 1 ; then observing that — cos a cos a' + 
sin A sin a' = — cos ( a+a') and that cos d + cos (o+a')= 2 
cos J {a+a'+d) cos J (a+a-^d) Art. 86, we have 



2cos^a+a'+rf)cos^a-f o'w ^) cosacosa* / t \n\ 

= ~~~ — ; — COS^A+Aj.^JLj 



COSD 

COS a COS a 



EXAMPLE. 

1. Suppose the apparent distance between the centres of 
the sun and moon to be 83^ 67' 33", the apparent altitude of 
the moon's centre 27^ 34' 5'', the apparent altitude of the 
sun's centre 48^ 27' 32", the true altitude of the moon's cen- 
tre 28° 20' 48', and the true altitude of the sun's centre 48^ 
26' 49'' ; then we have 

d = 830 57. 33"^ a = 270 34' 5", a' = 48° 2^ 32' 

A = 28° 20' 48", a' = 48° 26' 49" ; 

and the computation for D,by formula (1), is as follows: 



M 

176 APPLICATION OP TRIGONOMETRY. 

d 83057' 33" 

a 27 34 5 ar. comp. cos 052339 

a 48 27 32 ar. comp. cos 178383 



2) 169 69 10 


log. 2 301030 


|sum 79 69 35 
\ saxa^d 3 57 58 

A 28 20 48 
a' 48 26 49 


cos 9.239969 
cos 9.998959 
cos 9.944627 
cos 9.821719 



(Reject 40 from index) 1.536926= log .344292+ 



a+a' 76 47 37 nat. cos .228460— 

True distance S2P 20' 54" nat. cos .116832 



By glancing at the formula (1), we see that 30 must be re- 
jected from the sum of the above column of logarithms, to 
wit, 20 for the two ar. comp.' and 10 for r, which must be 
introduced into the denominator, in order to render the ex- 
pression homogeneous, so that the logarithmic line resulting 
ftom the process is 9.536926. Now, as in the table of log. 
sines, log. cosines, &c., the radius is supposed to be 10^^, 
of which the log. is 10, and in the table of natural 
sines, cosines, <fcc., the rad. is 1, of which the log. is ; it 
follows that when we wish to find, by help of a table of the lo- 
garithms of numbers, the natural trigonometrical line corres- 
ponding to any logarithmic one, we must diminish this latter 
by 10, and enter the table with the remainder. Hence the sum 
of the foregoing column of logarithms must be diminished 
by 40, and the remainder will be truly the logarithm of the 
natural number represented by the first term in the second 
member of the equation (1). If this natural number be less 
than nat. cos (a+a'), which is to be subtracted from it, the 
remainder will be negative, in which case d will be obtuse. 



NAUTICAL ASTRONOMY. 177 

Those who are desirous of .entering more at large into the 
problem of the longitude, and of becoming acquainted with 
the best methods of shortening the computation by the aid of 
subsidiary tables, may advantageously consult, besides the 
works already referred to, the Cluarto Tables of J. De Men- 
dozaRios, Lynn's Navigation Tables, Captain Eater's Treatise 
on Nautical Astronomy, in the Encyclopsdia Metripolitanai 
Kerrigan's Navigator's Ouide and Nautical Tables, and Dr. 
Myers's translation of Rossel on the Longitude. 

Variation of the Compass. 

114. We shall conclude this part of our subject by briefly 
considering the methods of finding the variation of the com- 
pass, or the quantity by which the north point, as shown by 
the compass, varies easterly or westerly from the north point 
of the horizon. 

The solution of this problem merely requires that we find 
by computation, or by some means independent of the com- 
pass, the bearing of a celestial object, that we observe the 
bearing by the compass, and then take the difference of the 
two. The problem resolves itself, therefore, into two cases, 
the object whose bearing is sought being either in the hori- 
zon or above it : in the one case we have to compute its 
amplitude^ and in the other its azimuth. 

The computation of the 
amplitude is simply deter- 
mining the hypothenuse of a 
right-angled triangle msn, 
of which one side is given, 
viz. the declination ns of 
the object, as also the angle 
opposite to it, viz. the colati- 
tude M. The computation 
of the azimuth requires the 
solution of an oblique sphe- 





178 APPLICATION OF TRIGONOMETRY. 

rical triangle, the three sides being given to find an angle ; 
the three given sides are the co- 
latitude pz, the zenith distance 
of the object zs and its polar 
distance ps : and the azimuth 
being measured by the angle I 
at the zenith z, opposite the 
polar distance, this is the an- 
gle sought. We shall give an # 
example in each of these cases 
of the problem. 

EXAMPLES. 

1. In January, 1830, at latitude 27^ 36' N., the rising am- 
jditude of Aldebaran was by the compass* K 23^ 30' N. ; 
required the variation. 

From the Nautical Almanac, it appears that the declination 
of Aldebaran at the given time was 16^ 9' 37" N., therefore 
since, by Napier's rule, Rad. xsin. dec. = sin. amp. x cos lat., 
the computation is as follows. 

sin. declination 16° 9' 37" . . . 9.444553 
COS. latitude 27 36 9.947533 



sm. amplitude E. 18 18 17 N. . . 9.497020 
Magnetic amplitude E. 23 30 N. 

Variation 5 11 43 



As the object is farther from the magnetic east than from 
the true east, the magnetic east has therefore advanced 
towards the south, and therefore the magnetic north towards 
the east; hence the variation is 6^ 11' 43" E. 



* The compass amplitude must be taken when the apparent altitude of the ob- 
ject is equal to the depression of the hoxiion. 



NAUTICAL ASTRONOMY. 



179 



2. In latitude 48^ 60' north, the true altitude of the sun's 
centre was 22^ 2' the declination at the time was 10^ 12' S., 
and its magnetic bearing 161^ 32' east. Required the varia- 
tion. 

O's polar distance 100^ 12' 
sin. zenith distance 67 68 ar. comp. 0.032936 
sin. colatitude 41 10 ar. comp. 181608 



2)209 20 

(S. 104 40 

sin. (l S. — zen. dist) 36 42 

sm. (i S. — colat.) 63 30 



. 9.776429 
. 9.961791 

2)19.942764 



sin. i azim. 69"" 26' 40" . . . 9.971382 



<S>'s true azimuth N. 138 61 20 E. 
observed azimuth N. 161 32 E. 



22 40 40 West. 



The variation is west, because the sun's observed distance 
from the north, measured easterly, being greater than its true 
distance, intimates that the north point of the compass has 
approached towards the west. 

3. In latitude 48^ 20' north, the star Rigel was observed to 
set 9° 50' to the northward of the west point of the compass ; 
required the variation, the declination of Rigel being S9 25' S. 

Variation, 22^ 33' West. 



i 



PART V. 



ADDENDA. 

116. We have postponed to this place the investigation of 
a few formulae requisite for the study of Analytical Geometry. 

By resuming the expression for the tangent, (Art. 32,) and 
patting a + 6 for a, we have 

R sin (a ± 6 

tan (a i: 6) = ^^« ,^ ^ tv 
^ ' cos (a ± 0) 

But by Art. 70 

. ^ , ,^ sin a cos 6 + sin 6 cos a 
sm (a ± 6) = ^= 

and _ 

, ,, cos a cos 6 4. sin a sin 6 
cos (a + 6) = -i 

substituting these values, the first equation becomes 

, , ^ R (sin a cos 6 + sin 6 cos a) 

tan (o + 6) = — ^ , -^ . ; — f-- 

^ — ^'^ cos a cos 6 4- sm a sm b 

dividing both numerator and denominator of the second mem- 
ber of this equation by cos a cos 6, we have 

/sin a sin b\ 

^ . , , * Vcos a'^cosbf 
tan (a + b) = = — v 

1 + , 

• cos a cost 

sin a sin b tan a 

substituting for -— - and -— r their values (Art. 32) 

tan 6 
' and -^ — - the last expression becomes 



QkNEliJLL FORMULiB. 181 

(tan a 4: tan h\ 

_tanatan6 
1 +~k3 



bfi 



, , a' (tan o + tan 6) 
taa(«±6)= ^,^^tan-^tan6 ^ ' ^ 

i. e. ^Ae tangent of the sum or difference qf two arcs is equal 

to rod. square into the sum or d^erence of their tangents 

divided byrad, square mi?ius or plus the rectangle of their 

tangents. 

If a represent the tangent of a and a' the tangent of a'^ then 

R being 1, 

a — a' 

tan («-«') = jq::^ 

Using the upper sines and making k == a inequation (1}| 
we have 

2 r8 tan a 
^^^ 2« = i«l4Sx^ • • (2) 

Tan 3a, tan 4a, ^c, may be foiihd by niaking b sueces^ 
sively equal to 2a, 3a, &c. 

116. The sine and cosine of 45^ are equal, since the com- 
plement of 45*^ is 45^. These two lines form two sides of a 
square of whibh r is the diagonal. Biit (Geom., B. 4, Prop. 
11, Cor. 4,) the diagonal is to the side of a square as v2 to 1, 
hence 

sin 46° = cos 45^ = --^ 

V2 

Multiplying both numerator and denominator by v2j thii 
value may be changed into 

sin 46° = i r Vg" 

117. The sine of ^ a7i arc is equal to J the chord ofth4 
arc. 



2r 



182 



AODKNOA. 



For let MN be the arc ; 
draw the diameter ba per- 
pendicular to the chord mn 
of this arc ; this perpendicu- 
lar bisects the chord, and j 
also the arc subtended by it ; 
(Geom., B. 3, Prop. 6,) but 
MP half the chord is the sine 
of MA half the arc, since mp 
18 a perpendicular from one 
extremity m to the diameter 
which passes through the other extremity a. 

ChroUary. The chord of 60°, or i of the circumference 
which is the side of the regular hexagon, is equsi to r 
(Geom., B. 6, Prop. 4,) j hence the sine of 30° is equal to J r* 

118. Referring to Art. 33, it will be observed that 




,3 



sec. = 



COS 



but, 
hence, 



COS 60° = sin^30° = i k 



R3 



sec 60° = J — =: 3 R = /Ae diameter of the circle^ 
a * 

119. Making d =r 46° in equation (1) of Art. 115, we 

have, R being 1, 

1 + tan 6 

tan (46° ±b)= -— • 

1 -|. tan 6 

120. Our demonstration for the sine and cosine of the sum 
of two arcs at Art. 69, might seem to want generality, since 
the arcs a and 6 are there supposed to be less than 90°. That 

hese arcs may extend to the other quadrants, can be shown 
iS follows : 
Let a=90°+w, then will the formula 

sin (a+6)=s sin a cos b+sin ft cos g . . . (1) 

R ^ 



♦ Tan 450 = cot 45o=s R. 



GENERAL FORMULAS. 183 

Still be true, for substituting 90^+m lor a, we have sin (90^ 
+fn+b) in place of the first member, which is equal to cos 
(m-fft)* ; for the second member by the same substitutioDy 
we have 

sin (90Q+m) cos 6+sin b cos (90+m) 

R 

but sin {90^+m) = cos m and cos (9(P+w) = — sin Mi 
hence equation (1) becomes 

cos (m+b) = cos m cos 6 — sin m sin 6 

which, since m and 6 are less than 90°, we know to be true, 
by Art. 69; hence (1), from which it is derived, is true also. 

Assuming (1) to be true with a > 9(P, which we have just 
proved, make 6==90o+w, and in a similar manner the truth 
of the formula may be established on the supposition of both 
a and 6 > 9(P. 

Afterwards make a = 180°-f m and observe that sin (180° 
4-w+6) = sin m+b and cos (180°+f»)= — cos m, and 
you will show that the formula extends to the third quadrant^ 
and so on. 

121. As we were not sufficiently advanced in the theory 
of trigonometrical lines to explain the construction of the 
tables of sines, tangents, &c., at Art. 38, having once been 
obliged to use them without explanation, we thought best to 
defer that to this place, so as not to interrupt, more than was 
necessary, the train of reasoning relative to the scdution of 
triangles. 

The diameter of a circle being multiplied by 7r=3.1415926 
we have the length of its circumference ; this divided by 
360 gives us the length of one degree, and this by 60 the 
length of one minute of the circumference. So small an are 
as V may be considered as equal to its sine, without sensible 
error. Having thus found the sine of I' we may find the 
sines of other arcs by formula (1), of Art. 74. 



* By referring to either of the diagrams in which a sine is drawn, it will I 
evident that sin(90<>-|-a),a being any arc less than a quadrant, is equal in lenf 
to siq (90<>— a) = cos a. AUo that cos (OO^+a) • — cos (9a<»—r' 



\ 



184 ADDENDA. 

2 

sin (a+6) + sin (a— 6) = — sin a cos A 

R 

fpaking a and b each equal to 1', this becomes 

sin ?' = — sin 1' cos 1' 

R 

observing that cos 1' = v r^ — sin^ 1' (Art. 72) ; we have 
thus the value of the sin 2'. Making a^=3f and ft=l^, we 
have, transposing sin (a — i), 

2 
sin 3' = — sin 2' cos 1' — sin 1' 

R 

Thus we find ^in 3' in terms of sin 2' before determined, 
-fpid so on. 

The cosines are calculated from fhe sines by the fbrmuli^ 



cos = vr* — sin* 

the tangents by the formula (Art. 32.) 

l^sia 

tan= 

cos 

fhe cotangents by (Art. 34) 

RCOQ 

cot = — ^— 
sm 

The sines, cosines, d&c, may be calculated by series, a 
specimen of which is given at Prob. iii., Art. 123. These 
series are most conveniently derived by the aid of the Differ- 
ential Calculus, and as those of our readers, who will wish to 
investigate them, are likely to become acquainted with the 
mode, in the study of that branch of Analysis, we have 
taken the liberty of quoting one of them i^t the article ahov^ 
^Qtioned. 



PART VI. 



MISCELLANEOUS TRIGONOMETRICAL INQUIRIES, 



122. We now come to the final part of our subject, in 
which we propose to bring together several miscellaneous 
particulars which properly come under consideration in a 
treatise on Trigonometry. Some of these might haye 
been introduced much earlier, but we have preferred to 
le^ve their consideration for a supplementary chapter, 
agreeing with Woodhotise, that it is better for the stu- 
dent first << to attend solely to the general solutions, and to 
postpone to a time of leisure and of acquired knowledge the 
consideration of the methods that are either more expeditious 
or are adapted to particular exigencies." 



CHAPTER I, 

ON THE SOLUTIONS OP CERTAIN CASES OP 
PLANE TRIANGLES, AND ON DETERMINING 
THE TRIGONOMETRICAL LINES OF SMALL 
ARCS. 

PROBLEM I. 

123. Given two sides and the included anglo of a pi 



186 MISCELLANEOUS INQUIRIES. 

triangle, to determine the third side, without finding the re- 
maining angles. 

The general expression for the side c, in terms of the two 
sides a, b, and the included angle c, is (Art. 68) on the sup- 
position of R = 1, 

c2 = a' -f 6^ — 2 a&'cos c 
= (a— 6)3+2 a6(l— cose) 
= la — bY + 2ab,2sin^^c 

4 aft . . 

= {a-bf {1 + (^Z6)5 ^^^-^ 4 ^ } 

Assume the second term within the brackets equal to tan.s^, 
then, since I + tan.^ = sqcJ^O = — ,^ , we have 



c = (a — b) 



rad 



cos 

Hence c is determined by these two formulas, viz., 

log tan *=log 2+4 log a+J log 6+log sin J c— log (a—b) 
log c=log (a — 6)+10 — ^log cos 

EXAMPLE. 

Given a = 562, b = 320, and c = 128° 4', to find c. 

log. 2 0.301030 

i log. 662 1.374868 

i log, 320 1.252575 

log. sin 64P 2' 9.953783 

ar. comp. log. 242 7.616185 log. 242+10 ; . 12,383816 

log. tan 6 10.498441 .-. log. cos 6 = 9.480718 

log. c 800.01 , . . 3.903097 

PROBLEM II. 

To determine the area of a plane triangle when any three 
Is except the three angles are given. 

• Let two sides, a, c, and the included angle b, be given. 

) fig. Art. 64.) 



PLANE TRIANGLESi 18t 

The area of the triangle is expressed by ^ bc . ad ; but ad 
t= ABsin. B ; hence the expression for the area, in terms of 
the given quantities, is 

area = ^ a o sin b 

2. Let two angles, a, b, and the interjacent side c, be giveii« 
Then, since 

sin c : sin A : : c : a^ 
we have 

sin A sin A sin B . 

a = ^: c .'*ac sin b=s 1 ^c* J 

sin c sm c 

hence the expression for the area is 

sin A sin b 

area = ^ . c^. 

2 sine 

3. Let the three sides be given. 
By Art. 73 

sin i r.—\/ ii^—a)ihs-c) 

ac 

Also^by adding (5) and (6) of Art. 72, and proceeding a» 
for the above, may be found. (See Art. 129.) 

cosiB= VHEE) 

ac 



.'. 2sin^BCOs^B,or(Art.71)sinB= — ^^s{is — ft)(i*— a)(i»— c)* 

dC 

Gonsequently, by substituting this value of sin b in the first 
expression, we have 

area= V^ s {^s — a) {^s — 6) (i*— ^) ; . 

which formula furnishes the well known rule, given in all 
books on mensuration, for the area of a triangle when the 
three sides are given. 

These expressions for the area of a plane triangle ate aB 
adapted to logarithmic computation. 



1S8 MtSCfcLLANfibrs ifrciufRiE0l; 

PROBLEM III. 

'To find the logarithmic sine of a very small arci 
By Article 74 Dif. Cal. (Davies') the expression for the sine 
of any arc x is 

sin x = x-y:^+ 1. 2 .3 .4 .5 -'^' 

Now, the length of an arc o( one degree is so small that, 
even when x is so great as this, the third term of the above 
iteiries can have no significant figure in the first ten places of 
tlecimals. 

Retaining therefore only the first two terms j we have, when 
or is small, 

*^'"=*-o:3='^^-2:3)=Ji— 2+2X4^ "^^y5 

that is, since the quantity within the brackets = cos z. (Dif. 
Calc, Art. 75.) 

sin. xr=z z cos.i x ; 

h^nce, by introducing the radius so as to render the expres- 
sion homc^eneous, 

log. sin a? = log. re — i (10 — log. cos x) ... (1) 
Let the arc x contain n secdnds, then 

* "^180^60x60 ^ 

hence, by introducing the radius, 

log. ic =± log. n + \6g. 3.14159, &c., + lO — log. 180 X 603 

= log. n + 4.685575 ; 

therefore, from (1), 

log. sin. a;=log. n+4.685575 — \ ar. domp. log. cos a? . . (2) ; 

hence this rule. To the logarithm of the arc reduced into 
seconds, with the decimal annexed, add the constant quantity 
4.685575, and from the sum subtract one third of the arith- 
metical complement of the log. eosine ; the remainder will be 
the logarithmic sine of the given arc. 

This rule will determine the log. sine of a very small arc 
I great accuracy; it was first given, without demon- 



SAtALt ARCBi 189 

MtatioD^ by Dr. Maskelyne, in his Introduction to Taylor's 
Logarithms^ The abo^e proof is from Woodhouse's Trigo- 
nometry. 

PROBLEM IVi 

To find the logarithmic tangent of a very small arc. 

Let X be the arc ; then, as we have found in last problem 

i sin a? ^ X 

sm a; = z cos tt .-. = tan x =i 



cos Z C0S3 x* 

tlence, introducing the radius, 

log tan a; = log a: + J (10 — log cos z) 

The second member of this equation is equal to the second 
member of (1) in last problem, plus the arithmetical comple- 
ment of log cos a: ; hence, since the second member of (2) is 
equivalent to the second member of (1), we have 

Iogtana;=Elogn+4.686575-|-f arith. comp. log cosa? . . (3)J 

which furnishes this rule. To the logarithm of the ar6 
reduced to seconds add the constant quantity 4.685576, and 
two thirds of the arithmetical complemetitof the log. cOsinei 
the sum is the log tangent of the given arc. 

t>ROBLEM Vt 

To find a small arc from its log» sine or its Idg; tai)gent. 
1. Let the log. sine be given ; then n^ being the number ot 
seconds in the arc, the expression (2), in problem iii., gives 

log. n= log. sin x — 4-685676 ++arith.comp.log. cos x* 
= log. sin ar+6.31442d — 10+iarith. comp. log.cos Ss 

therefore, to find the arc from the log. sine the riile is thiit. 
To the log. sine of the small arc add 5.314425, and i of 
the arithmetical complement of the log. cosine ; subtract 10 
frdm the index of the sum, and the remainder- will be the 
logarithm of the number of seconds in the arc. 

8. Let the log. tangent be given ; then from the ekprei* 
MOD (3), last problemi we have 



190 MI8CfiLL4II£0XI9 INCIUIRIES. 

log. n=log.ian ar — 4.686676 — ^arith.eoinp.log.ca8 1. 

ssslog.tan X + 5.31442a — 10 — |arith.comp. log. eos r ; 

that is, to the log. tangent of the small arc add 6.414426, and 
from the sum subtract f of the arithmetical complement of 
the log. cosine, take 10 from the index of the remainder, and 
we shall haVe the logarithm of the number of seconds in 
the arcr 

PROBLEM vi.. 

A person on one side of a 
river observes an obelisk on 
the opposite side, and, being 
desirous to ascertain its height^ 
he todL with a quadrant the 
angle b = 66^ 64', which the 
obelisk subtended at the place 
where he stood, then going back the distance bas= lOOfeet^ 
lie again measured the subtended angle, and found it to be 
4 ^:;= 33^ 20' ; what was the height ef the obelisk 7 

The problem may be solved readily, as follows ; 

If we take cd far radius, db will be the tangent of the 
angle dcb, and da the tangent of dca, therefore, ab is the 
difference of those tangents ; but by referring to the table of 
natural sines and cosines, we find that to radius 1 

nat. tan 56<> 40' = 1.520426 
nat. Ian 34^6' = 677061 




difibreneess .843876 



.-. .8433762 : 1 : : 100 : 118.67 

FR09LBM VII. 

^uprp^ee that front the tt^ of a moantain, three miles 

h^h, the angl^e pf depreasknn of the remotest visible point 

^liMieartti^ wrfetee ia taken and found to be 2^ 13' 27" ; 

required thence to determine the diameter of the eartb| 

XNnngit to Idi a perfect sphere* 



WhAXK TRIANOLSfl. 



191 



Let o be the centre of the eftrth, 
BA the mountain, ac the visual ray 
or line touching the earth's surface 
in c. Draw the tangent bd^ and 
join on, oc ; then the angle of de* 
pression eac being given, we have 
also the angle bad, the complement 
of it, equal to 87° 46' 33". Also since 
the tangents bd, cd, are equal, 
(Geom. p. 64,) we have the angle 
BOD = DOC =:: ^ comp. A = 1° 6' 43^ and,, therefore^ 900 
= 88° 53' 16i", 

Now in the right-angled triangle abd we have 

BD = AB tan A ; 
and in the right-angled triangle obd, 

OB = BD tan boo ; 
hence, by substitution, 

OB = AB tan A tan bdo ; 
the computation is, therefore, as follows : 




AB=: 


' t> « • • • 


0.477121 


tan A 


870 46' 33" 


. 11.410738 


tan BDO 


88° 53' 16J" 


11.711931 



OB 



3979.16 



3.699790 



henoe the diameter is 7958.3 miles. 



PROBLEM VIII. 



Given the distances between three objects a, b, c, and the 
angles subtended by these distances at a point d in the same 
plane with them ; to determine the distance of d from each 
object. 



4 



IW 



MISCJSLLaifEOUS INaUIBlSS. 





Let a circle be described about 
the triangle adb, and join ae, eb, 
then will the angles abe, bae, be 
respectively equal to the given 
angles ape, bpe, (Qeom., p. 66,) 
thus all the angles of the triangle 
aeb are known, as also the side 
AB ; we may find, therefore, the 
remaining sides ae, eb. Again^ 
the sides of the triangle abc being 
known, we may find the angle 
BAG ; hence the angle cae be- 
comes known, so that in the tri- 
imgle CAE we shall have the two 
sides AE, AC, and the included 
angle given, from which we may 
^nd the angle aec in fig. 1, or the 
(Uigle ace in fig, 2, and thence its 
supplement aed or acd; this with 
the given side ae and angle ade, in the first figure, or with 
the given side ac and angle adc in the second, will enable 
us to find AD, one of the required lines, and thence dc and 
PB the other two. 

Qr the solution may \)e conducted more analytically as 
follows ; 

Put X for the angle dac, and x' for the angle dbc ; ajso pall 
the given angles adc, bdc, a and a then a, 6, c, representing 
fis usual the sides opposite to a, b, c, we have 

sin a ^ sin u _a^ 

sm X PC siq a;' PC \ / 

sin a sin a;' b , , 

"•^tt: -TT;;r =^ — •'• « ^^^ " sin x' = b sm «' sm a: . , (2) 
sm a sm X a ' v / 

This is one eqqatiou bet>veen the unknown quantities x, x' 

Another is easily obtained ; for since the four angles of the 

quacir I lateral adbc make up four right angles or 360°, we 

T + a' + a + a' + ACD + BCD = 360° ; the two latter 

oqAy be pOQsidered as l^r^own, sinp« in the triangle abc 



PLANE TRIANGLES. 193 

the angle c is determinable from the three given sides ; there- 
fore all the terms in the first member of this equation are 
known except x and x\ Call the sum of the known quan- 
tities ft and we shall thus have re* = (9 — a:, and, conse« 
quently, by substitution, equation (2) becomes 

a sin a sin (i? — a; ) = 6 sin o! sin x 

= a sin a (sin l? cos a; — cos j? sin %) ; 

or dividing by sin x, 

j^ 3iu a' = a sin a (sin /? cot a; — cos (9) 

b sin a' cos ^ 

,'• cot a;== :: : — tA — 2> 

a sm a sm (? ' sm j? 

^ sin a' , 

= -• J — Tfvr cot p. 

a sin a sin p ^ 

The first term of this second member may be easily calcu* 
lated by logarithms, and this added to the natural cotangent 
of ^ gives the nat. cot. of «, and thence x is known from the 
equation x'=^^'. — a;, and cd fi'om either of the equations (1). 

This problem has a useful application in the survey of 
harbors, 

Let the angles be taken with a sextant, from a boat, at a 
point where a sounding is made, to three stations on the 
3hore, After having drawn upon a map the triangle, of 
which these three stations are the vertices, the following 
simple and elegant construction will determine the point 
where the sounding was made. 

Upon the line joining two of the stations, on the map, 
make a segment, capable of containing the angle observed 
from the place of sounding, and subtended by this line (Geom., 
B. 3, Prob. 16) ; upon a line joining one of these two sta- 
tions and the third, make another segment, that will con- 
tain the angle observed to be subtended by this last line, and 
the intersection of the arcs of these two segments will deter- 
mine the point on the map, corresponding to that at which 
the sounding was made. 

PROBLEM. IX. 

Qiveiii the angles of elevation of an object taken at three 



194 



MISCELLANEOUS IN<%UIBIES. 




places on the same horizontal straight line, together with the 
distances between the stations ; to find the height of the ob- 
ject and its distance from either station. 

Let AB be the object, and c, c', 
c", the three stations, then the tri- 
angles BcA, bc'a, bc"a, will all be 
right angled at a ; and, therefore, to 
radius ba, ac, ac', ac", will be the 
tangents of the angles at b, or the 
cotangents of the angles of eleva- 
tion ; hence, putting a, a', a", for 
the angles of eleFation, x for the 
height of the object, and a, b^ for 
the distances c &, d c", we shall 
have 

AC=a; cot a, Ac'=a: cot a', Ac"=a; cot a" 

Now if a perpendicular ap be drawn from a to cc", we 
shall have (Geom., p. 80,) from the triangle acg' 

AC«=Ac'«+c'c«— 2 c'c . c p ; 
and from the triangle ac'c" 

AC"«=AC'3+C"c^+2 c"c . c'p ; 

that iS| we shall have the two equations 

3^ C0t3 a =5=2:2 coi2 a'+a»-^2a . c 'V 
x'^cot^ a''=:x^COi^ a'+b^+2b . c'p. 

9 

in order to eliminate c'p, multiply the first by 6, the second 
by a, and add, and we shall have 

a^b cot2 a+a cot^ a'')z=z{a+b) s^ cot^ a'+ab (a+b) 



/. x= \/- 



ab (a+b) 



b cot^a-l-a cot«a''— (a+6)cot« «'* * 

If the three stations are equidistant, then a=6, and the ex- 
pression becomes 

a 

«== -7=======================^ 

^ i cot» o+i cot« a"— -cot2 a ' 



SPHERICAL TRIANGLES. 



195 



The height ab being thus determiDed, the distances of the 
stations from the object are &und by multiplying this height 
by the cotangents of the angles of elevation. 

Solution of Quadrantal Triangles. 

(124.) The rules for right-angled triangles will serve also 
for the solution of quadrantai triangles^ or those in which 
one side is a quadrant. For by changing such a triangle for 
its supplemental or polar triangle, we shall then have to con- 
sider a right-angled triangle, of which the hypothenuse will 
be the supplement of the angle opposite the quadrantai sidOj 
the two perpendicular sides supplements of the other ■ two 
angles of the proposed triangle, and the two oblique angles 
of the new triangle supplements of the oblique sides of the 
primitive triangle. That is, the sides of the primitive or 
quadrantai triangle being a, 6, and c = 90°, and its angles A| 
B| c, the sides of the supplemental triangle will be 180°j — Af 
180° — B, and 180° — c, this latter being the hypothenuse ; and 
the opposite angles wiQ be 180°— a, 180^—4, and 90°. But 
the parts of a quadrantai triangle may be determined with- 
out the aid of the supplemental triangle. 
Thus let AD be the quadrantai side in the 
triangle abd. Produce db, if necessary, 
till DC becomes a quadrant, and draw the 
the arc ac, which will, obviously, measure 
the angle d, since d will be the pole of the 
arc AC, and c will be a right-angle : also 
the angle cab will be the complement of 
the angle bad in the proposed triangle, and 
the angle abc will either be identical with 
abd in the proposed, or supplemental to it, 
accordingly as dc exceeds, or falls short of, a quadrant; 
hence all the parts of the proposed triangle are easily dot' 
mined from those of the right-angled triangle abc. 

If the angle dab is less than 90°, or than the angle 1 
the side db must, obviously, be acute ; but if dah is 
than 90°, db will be obtuse, and converael) 




id6 MISCELLANEOUa INdtJIRIEIJ. 

angles tidjacient to the quadrantal side are of the same npQ* 
cies as the sides opposite to them; The same may be infei'^ 
red from the polar triangle^ 

It must be remarked, that the solution will be ambiguous 
Whenever the determination of the right-angled triangle be^^ 
comes ambiguous, whether we employ the polar triangle, or 
the triangle abc in the above diagram. *This ambiguity 
occurs only when the given parts in the right-angled triangle 
are one of the perpendicular sides, and the opposite angle to iti 

EXAMPLES. 

In the triangle dAb, da = 90° a = 64^ 43', and d = 42^ 
12' required the other parts* 

As the angle dai^ is less than 
dO^, that is less than the an-^ 
gle DAC, DB is less than a 
quadrant, and, therefore, the 
tight-angled triangle abc is 
sittia{ed as in the figure, bc 
being the prolongation of db. 
Of the parts of this right-an- 
gled triangle we have given a 
= 900—640 43' =t: 350 17', and 
6 r= 420 12', to find the other 

parts. 

Let A be the middle part, then b and c will be adjacent 
parts, therefore, 

rad< X sin. comp. a :^ tati. b x tati. cotnp. (f, 
that is, 

tod* COS. A 




raa« cos« 


A — lan. c( 


51. C .\ C 


01. c = - 


tan. b 


rad. . 


. • 


. * 


• 


10.000000 


COS. A 


350 ir 


« 


• i 


9.9U863 


an. b 


42 12 


. . 


. 


. 9.967486 



»t. c 48° 0' 9" . . . 9.964368 



SPHERICAL TRIANGLES. 197 

Xiet B be the middle part) then a, 6, will be opposite 
parts, and, consequently, 

rad. X sin comp. b = cos 6 x cos comp. a ; 

that is, 

, cos b sin A 

rad cos B = cos 6 sm a ••. cos b=c 3 — 

rad. 

rad . . . 4 • . 10.000000 
cos 6 42^12' . . . 9.869704 

sin A 35 17 . * • • 9.761642 



cosB 64<5 39'56" . . . 9.631346; 



hence the angle abd is 115^ 20' 5''. 

It remains now to find a ; let, therefore, b be the middle 
part, then a and c will be the adjacent parts ; hence 

rad X sin comp. b = tan a x tan comp. c ; 
that is, 

, * xx^ tad cos S 

rad cos b = tan a cot c .*. tan a = — ^ 

cot c 

rad 10.000000 

cos b ....*. 9.631346 

cote . : . . 9.964368 



tana 25^25' 20" . . . 9.676978 



therefore, the side db, which is the complement of thiS| is 
64° 34' 40.' 



d9 



198 MX8CBI.LANB0US INi^UIRIES. 

2. In the triangle dab, da = 9(P, a = 112° Z 9", anc 
AB = 67^ 3' 14", to find the other parts. 

Since in this example a is obtuse, db 
is obtuse. 

In the right-angled triangle abc we 
have A = 220 2 9" and ab = 67° 3' 14" ; 
let A be the middle part, then ab, ac, will 
be adjacent parts, and we shall have 

rad xsin comp. A=tan b x tan comp. c; 

that is. 




rad cos A = tan 6 cot c 


rad cos a 

/. tan ft = 1 — 

cot c 


rad .... 


10.000000 


cos A 22^2 9' 


. 9.967056 


cot c 67 3 14 


9.626716 


tan 6 66 27 9 


10.340341 



therefore, the angle d = 65° 27' 9". 

Take now a for the middle part, then a and c will be oppo- 
site parts ; hence 

rad X sin a = cos comp. a x cos comp. c, 
that Is, 

. . sin A sin c 
rad sin a = sm a sm c .\ sm a == — —g 

and a will be acute, because the opposite angle is acute 

rad 10.000000 

sin A 22^2 9'' ... 9.574247 

sine 67 3 14 . . . 9.964199 



sin a 20 12 44 . . . 9.538446; 



therefore bd = 110° 12 44". 

As we have now to find b, take a for the middle part, then 
6 and b will be adjacent parts, therefore 



SPHERICAL TRIANGLES. 199 

rad X sin a = tan fr tan comp b ; 
that is, 

, . rad sin a 

rad sin a=tan b cot b .•. cot b=-: — j — 

tan b 

rad. . . . 10.000000 

sin a . . . . 9.538446 

tan ft . . . 10.340341 



cotB 8P1'58'' . • 9.198106 



125. Napier's rules for the solution of right angled spheri- 
cal triangles, though applicable to all cases, do not give re- 
sults of that degree of accuracy, which is sometimes required, 
when the sought part expressed by its sine is very small, or 
expressed by its cosine, is very near 00°. The following 
formulae may in such cases be used. 

I. 

At Art. 86, by the formula which follows, (6), we have, 
R being 1, 

1 — cos p = 2 sin^ J p 

in a similar manner, we might obtain 

l+cosp = 2 cos^ \p 
whence 

1 — COS/? 

j-p^=tan«4p 
But by Napier's rules, r being 1, • 

cos a = cot B cot c 

changing p into a, and substituting the value of cos a, given 
by this last, we have 

^ „ 1 1 — cot B cot c sin B sin c — cos b cos c 

tan-* A a =. , -. ^ ^^- : ; 

^ 1 + cot B cot c sm B sm c + cos b cos c 

or ^ , cos(b+c) 

tana i a = -7 — —i 

^ cos (b — c) 



000 MXICSLLANE0U8 IKdVIRIES. 

which is a formula to be employedi when b and c are given 
mi a leciuiied, 

II. 
With the same data to find b use the formula 

tan i 6 = v| tan [^(b— c)+45o] tan [i(B+c)-46o] \ 
derived from 

cos B 

COS & =^g- ■ and formula (6), Art. 74, 

III. 

The hypothenuse a and the side c being given to find the 
adjacent angle b, use the formula 



Wsin(a— e) 
tan i b = V -— (^^) 

derived .in a manner similar to that in Case I, 

ly. 
By Napier's rules 

cos a 



cos c =rrrT^ 
cos 

whence 

tan 4 c = tan J {a+b) tan ^ (a — b) 

V. 

Finally, to obtain b when the opposite angle b and the 
hypothenuse a are given, we have, by Napier's rules, 

sin b z;=sina sin b 

1 — tan X 
whence, observing that i^^^ ^ = tan (45°— a:) (Art. 119) 

i making tan x = sin a sin b, 



tan (460— J*) = ^ tan 45^— » 



•PH£RICAL TAIANGLBS. 301 

This last gives the value of 6, x being calculated by the 
equation tan x = sin a sin b. 

N. B. In the above five formulae, r is supposed to be 1. To 
introduce it correctly when logarithms are applied to the for- 
mulae, it is only necessary to observe that the two members of 
each must be homogeneous. 

126. When, in the case considered at Art. 87, the only 
part rec[uired happens to be the side opposite the given angle, 
the finding of the other two angles then becomes merely a 
subsidiary operation, and the determination of the required 
side, by Napier's analogies, seems somewhat lengthy. But a 
shorter method of solution is deducible from the fundamen- 
tal formula, 

cos c = cos a cos ft + sin a sin 6 cos c (1). 

For substituting cos a tan a for its equal sin a it becomes 

cos c = ccs a (cos b + tan a sin b cos c). 

Assume 

cos 61 

tan a cos c = cot w = — : — ; 

smw' 

then 

sin 01 cos 6 + sin 6 cos « 



cos c = cos a- 



sm (o 



cos a sin {(a + b) , 

sin oi 

Hence, to find the side c, we must determine a subsidiary 
angle a from the equation 

cot oi = tan a cos c (2) ; 

after which c is found by the equation 

cos a sin (w + 6) 

cos c= r-^^ ^ .... (3). 

smw ^ ' 

EXAMPLE. 

1 . In a spherical triangle are given a = 38^ 30', b = 7(P, 
and c = 3P 34' 26", to find c. 



202 MISCELLANEOUS IVaVIRlES. 

tan a 380 30' q" . . 9.900605 cos a . . . 9.893544 
cos c 31 34 26 . . 9.930422, sin w, ar comp. 0.082065 



cot « 55 52 30i . . 9.831027, sin (w + ft) 9.908644 



cos c 40° . 9.884253 



Other formulas for the determination of c might be easily 
deduced from the same equation (1), but this is as short and 
as convenient as any. We might also introduce here a dis- 
tinct formula for the determination of one of the angles a, 
by help of a subsidiary arc w ; but as little or nothing would 
be gained, in point of brevity, over the process by Napier's 
analogies, we shall not stop to investigate it. 

127. If where two angles and the included side are given, 
the angle opposite to the given side be the only part required, 
a more compendious method of solution may be obtained by 
introducing a subsidiary arc, as in the last case. Thus from 
the fundamental formula for the cosine of a side in terms of 
the three angles, might be obtained by aid of the polar trian- 
gles, a formula which becomes when cos a tan a is sub- 
stituted for sin a, 

cos c = cos A (tan a sin b cos c — cos b) ; 
or assuming 

cos 0) 

tan A cos c = cot w : 



sm 6i 
sin B cos w — sin w cos b 



cos c= cos A 



sm » 

cos A sin (b — w) 
sin (9 ^ 

Hence, having found a subsidiary angle w, by the equation 

cot fti = tan A cos c .... (1); 
the sought angle is determined by the equation 

cos A sin (b — w) 

cos c ==^ = 

r smw 



SPHERICAL TRIANGLES. 203 

EXAMPLE. 

In a spherical triangle are given two angles equal to 39° 
23' and 33° 45' 3 " and the interjacent side 68° 46' 2" to find 
the third angle, 

tan A 39° 23' 0" 9.914302 cos a . . 9.888133 

cos c 68 46 2 9.558898 sin oi, ar. comp. . 0.018392 



cot oi 73 26 33J 9.473200sin(B-HM)39°41'30i" 9.805268 

cos c 59° 0' 13' 9.71 1793 



As (b — oi) is negative, cos c must be negative ; hence c is 
the supplement of this, viz. 120° 59' 47". 

123. The part of a spherical triangle determined by the 
proportion sin a : sin 6 : : sin a : sin b admits of a double 
value, since two arcs answer to the same sine ; it becomes 
necessary, therefore, for us to inquire under what circum* 
stances both these values are admissible, and how we may 
know which to choose when but one solution exists. 

Referring to the fundamental formula (Art. 82,) we have 

cos b — cos a cos c 

cosB = * • ; . . . (2) 

sm a sm c ' ^ ' 

in which expression wc may remark that if cos b is numeri- 
cally greater than either cos a or cos c, the second member 
must take the sine of cos 6, consequently, b and b must be of 
the same species if sin & < sin a , or sin b < sin c, that is, an 
angle must be of the same species a^ its opposite side, if the 
sine of this side is less than the sine of either of the other 
sides. 

But if cos b is numerically less than cos a, then whether 
the right hand member be + or — will depend upon the mag- 
nitude of cos c, or cos c will have two values corresponding 
to -f cos B, and — cosb; hence an angle has two values, 
when the sine of its opposite side is greater than the sine 
of the other given side. 

In the proportion 

sin A : sin b : : sin a : sin b 



804 MI80ILL1NE0U8 INQUIRIES. 

b being the required part, the nature of the arc b may be dis- 
cussed, as in the preceding case. 

By means of the polar triangles, we obtain from (2), in the 
same manner as at Art. 85, the formula 

cos B -I- cos A cos C 

cos b = : ' 

sm A sm c 

from which it follows, as in the foregoing case, that if cos b 
is numerically greater than cos a, b and b will be of the same 
species. If cos b is numerically less than cos a, then both 
the values of 6, given by the above proportion, will be admis- 
sible, for c may be determined so as to render cos b positive 
or negative. Hence antf side will be of the same species as 
its apposite angle^ if the sine of this angle be less than the 
sine of either of the other angles ; and the species of the side 
b will be indeterminate if the sine of its opposite angle b 
be greater than the sine of the other given angle a. 

129. Adding instead of subtracting equations, (5) and (6) 
of Art. 72, we should obtain a formula for the cosine of half 
an arc in terms of the whole arc, and then proceeding in the 
same manner as at Art. 84, we should obtain a formula for 
the cosine of half an angle in terms of the three sides of a 
spherical triangle, viz., 



cos A a _^/ «ini^sin(^^-a ) 

sin b sin c 

in which r = 1. Dividing the formula for sin J a derived 
in Art. 84 by this, the result is 

taniA= V— ^M— ^^-'^'^^i-^"!-^-) 
^ sin ^ s sm {^s — a) 

Either of these applies when three sides of a spherical tri- 
angle are given. The formula of Art. 84 is best to use when 
the required angle is near 90^ ; the first formula of this arti- 
cle, cos ^ A = &c., when the required angle is very small • 
and the last in which the required angle is expressed by means 
of its tangent, when the angle is of intermediate magnitude. 



Et^^Edl* OP MiNVTB ERitORS. 20S 

By^ the aid of thd polar triangles we obtain 

Sin B sm c 
and, 

tan 



ia= -v/ ^Qs ^ s cos (^ s— a) 
* * cos ( Js -^b) cos (J s — c) 

f()]rmtilse which may be employed in a similar manner, whail 
the three angles of a spherical triangle are given. 

130 In practice, the data for the solution of triangles are 
obtained by observation and measurement^ and are liable to 
error from obvious and inevitable causes. It is true that from 
the great excellency of instruments, and the almost incon- 
ceivable accuracy of modern observation, these errors are 
extremely minute, yet, in cases where great precision is 
jfequisite, it becomes necessary to determine the effects which 
small errors in the data will produce upon the computed 
quantities, and to select the data and quaesita in such a manner, 
that the given errors in the one, shall entail the smallest pos- 
sible upon the other. The errors which are produced in 
determining the latitude and longitude by the neglect of cer* 
tain corrections noticed in the last part of Art. 110, are 
instances of the practical importance of such an investigation^ 

The principles of the Differential Calculus present an easy 
method for the purpose in question. 

Let lis suppose that of the three data (for there are al\)^ays 
three in the solution of a triangle) two have been obtained 
with sufficient accuracy, but the third x is liable to an erroi* 
of a given amount, which we shall call h^ Let u be the 
sought quantity^ Two of the three data being considered 
constant, the sought quantity u may be considered as a func-^ 
tion of the third z. The quantity x becoming x + h, let tho 
quantity tl become w'. We have by Taylor's theorem, (Davies' 
Dif. Calculus, Art. 42) 

du cPu (Pu 



30 



y!~^u is the error sought, and as A i3 in praetid) very small, 
(he higher powers of it may be neglected, and we may call 

}ieace the following rule ; 

Multiply the given error by the differential corofficient of the 
sought quantity considered as a function of the given ^uan- 
tity liable to error ^ and the product will be the error in the 
nought quantity. 

If two of the data be liable to given errors, the effect upon 
tl^e sought quantity may be computed on similar principles, 
\>Y considering the sought quantity as a function of the two 
data so liable to error, and differentiating it with respect to 
these as two independent variables, 

It is evident that the same method extends to the case 
'^here all the data are liable to given small errors. In this 
case the sought quantity is to be regarded as a function of 
three independent variables, and its di^erential found as 
l)efore, 

EXAMPLE, 

To determine the relation between the minute variations 
of the leg of a plane right angled triangle and the opposite 
angle, the remaining leg being considered Constant. 

Let A and a be the side and angle which are subject to vc^- 
riation, and h the constant side. Then, (Art, 41,) 

a = 6 tan A 
and (Dif. Cal., Art. 66) 

da 

which is the multiplier of the given small variation in a to 
obtain that in a. 

The theory of maxima and minima as explained in the 
Calculus, will here admit of an important and easy applica- 
tion, viz., to find under what circurnsrapces u* — u will be 



EFFECT OF MINUTE ERRORS. 207 

least on the supposition of a given variation h in the variable 

datum, or in other words, under what circumstances the 

du 
fpnction -t" wijl be a minimum, 



A TABLE 



LOGABITHMS OF HUMBEBS 

FBOB 1 to 10,000. 



15 

i 

33 

3B 


U.!. 


27 

28 
29 
30 
31 

3G 

3il 
40 
41 
43 

fi 

45 

47 
48 
49 
GO 


— 


Ux. 


51 
53 
S3 

64 
66 
66 
67 
58 

60 
6l 
63 
63 
64 

67 

70 
Tl 

73 
73 

76 


L^- 


78 
79 
80 
81 
83 

84 
87 
90 

95 

96 
97 
98 
99 
100 




Loit. 







000000 
301030 
177131 
603060 
698970 




414973 
431364 
447158 
482398 
477121 




724276 
733394 

740363 


- 


S808U 
886491 
892096 
897637 
903090 
"908486 
913814 
9I907S 




778151 
ei.'iOSS 
903Q90 
054343 
000000 




505150 
544068 




748188 
755S75 
763438 
770853 
778151 




011393 
079181 

146128 
1781)91 




658303 
56S202 
579734 
591065 
602060 




783393 
793341 
ROB 180 
813913 




93449S 

944483 
949390 
951213 




304180 
330449 
265S73 
278754 
301030 




812784 
623249 
633468 
643453 
653213 




819544 
836076 
83S509 
838849 
845098 




963788 
988483 
973138 
977724 




332319 
343423 
301728 
380211 
397940 


1 


663758 
673098 

690190 
69R970 




8512SS 
857333 
863323 
869232 
875061 




9S3371 
980772 
991236 
995635 
0000(10 



N. B. In the following tablet in (be last nine column! of 
each page, where the first or leading figures change from 9's 
to O's, points or dots are introduced instead of the O'a through 
the rest of the line, to catch the eye, and lo indicate that from 
(hence the annexed firat two figures of ihe Logarillim in the 
■econd coluina stand in the next lower line. 



A TABLE OF LOOARITIUfS FCOX 1 TO 10,000- 



N. 0ll|2|a 4i5i6 7i8|9|D. 1 


100 1 OOOOOOi 0434 


0868 


1301 1734|2166!2508| 


3029 


3461 


3891 


432 


101 


4321 


4751 


5181 


5609 


6038 


6466 


6894 


7321 


7748 


8174 


428 


102 


8660 


9026 


9451 


9876 


..300 


.734 


1147 


1570 


1993 


2415 


424 


103 


012837 


3259 3680 


4100 


4521 


4940 


6360 


6779 


6197 


6616 


419 


104 


7033 


745 L 


7868 


8234 


8700 


9116 


9532 


9947 


.361 


.776 


416 


105 


021189 


1603 


2016 


2428 


2841 


3252 


3664 


4076 


4486 


4896 


412 


106 


6306 


5715 


6125 


6533 


6942 


7350 


7757 


8164 


8571 


8978 


408 


107 


9384 


9789 


.195 


.600 


1004 


1408 


1812 


2216 


2619 


3021 


404 


108 


033^124 


3826 


4227 


4628 


5029 


5430 


5830 


6230 


6629 


7028 


400 


109 
110 


7426 


7825 
1787 


8223 
2182 


8620 
2576 


9017 
2969 


9414 
3362 


9811 
3755 


.207 


.602 
4540 


.998 
4932 


396 
393 


041393 


4148 


HI 


5323 


5714 


6105 


6495 


6885 


7275 


7664 


8053 


8442 


8830 


389 


112 


9218 


9606 


9993 


.380 


.766 


1153 


1638 


1924 


2309 


2694 


386 


113 


053078 


3463 


3846 


4230 


4613 


4996 


6378 


5760 


6142 


6524 


382 


114 


6905 


7286 


7666 


8046 


8426 


8805 


9185 


9663 


9942 


.320 


379 


115 


060698 


1075 


1452 


1829 


2206 


2582 


2958 


3333 


3709 


4083 


376 


116 


4458 


4832 


5206 


5580 


5953 


6326 


6699 


7071 


7443I 


7815 


372 


117 


8186 


8557 


8928 


9298 


9668 


..38 


.407 


.776 


1145 


1514 


369 


118 


071882 


2250 


2617 


2985 


3352 


3718 


4085 


4451 


4816 


6182 


366 


119 
120 


5547 


5912 
9543 


6276 
9904 


6640 
.266 


7004 
.626 


7368 
.987 


7731 
1347 


8094 
1707 


8467 


8819 
2426 


363 
360 


079181 


2067 


121 


082785 


3144 


3503 


3861 


4219 


4576 


4934 


6291 


5647 


60<)4 


357 


122 


6360 


6716 


7071 


7426 


7781 


8136 


8490 


8845 


9198 


9662 


355 


123 


9905 


.258 


.611 


.963 


1315 


1667 


2018 


2370 


2721 


3071 


351 


124 


093422 


3772 


4122 


4471 


4820 


5169 


5518 


6866 


6215 


6662 


349 


125 


6910 


7257 


7604 


7951 


8298 


6644 


8990 


9336 


9681 


..26 


346 


126 


100371 


0715 


1059 


1403 


1747 


2091 


2434 


2777 


3119 


3462 


343 


127 


3804 


4146 


4487 


4828 


5169 


5510 


6851 


6191 


6531 


6871 


340 


128 


7210 


7549 


7888 


8227 


8565 


8903 


9241 


9579 


9916 


.253 


338 


129 
130 


110590 


0926 
4277 


1263 
4611 


1599 
4944 


1934 

6278 


2270 
5611 


2605 
6943 


2940 
0276 


3276 
660i 


3609 
6940 


335 
333 


113943 


131 


7271 


7603 


7934 


8265 


8595 


8926 


9256 


9586 


9916 


.245 


330 


132 


120574 


0903 


1231 


1560 


1888 


2216 


2644 


2871 


3198 


3526 


328 


133 


3852 


4178 


4504 


4830 


5156 


5481 


6806 


6131 


6466 


6781 


325 


13t 


7105 


7429 


7753 


8076 


8399 


8722 


9045 


9368 


9690 


..12 


323 


135 


130334 


0655 


0977 


1298 


1619 


1939 


2260 


2580 


2900 


3219 


321 


136 


3539 


3858 


4177 


4-196 


4814 


5133 


.5451 


5769 


6086 


6403 


318 


137 


6721 


7037 


7354 


7671 


7987 


8303 


8618 


8934 


9249 


9564 


315 


138 


9879 


.194 


.508 


.822 


1136 


1450 


1703 


2076 


2389 


2702 


314 


139 
140 


143015 


3327 
6438 


3639 
6748 


3951 
7058 


4263 


4574 


4885 


5196 


5507 
8603 


5818 
8911 


311 
309 


146123 


7367 7676 


7985; 8294 


141 


9219 


9527 


9835 


.142 


.449 


.756 1063! 1370 


1676 


1982 


307 


142 


152288 


2594 


2900 


3205 


3510 


3815 


4120! 1424 


4728 


5032 


305 


143 


5336 


5640 


5943 


6246 


6549 


6852 


71.54 


7467 


7759 


8061 


303 


144 


8362 


8664 


8965 


9266 


9567 


9868 


.168 


.469 


.769 


1068 


301 


145 


161368 


1667 


1967 


2266 


2564 


2863 


3161 


3460 


3758 


4055 


299 


146 


4353 


4650 


4947 


524-1 


6541 


5838 


6134 6430 


6726 


7022 


297 


147 


7317 


7G13 


7908 


8203 


8497 


8792 


9086 


9380 


9674 


9968 2951 


148 


170262 


0555 


0848 


1141 


1434 


1726 2019 


2311 


2603 


2895 


293 


149 
150 


3186 


3478 
6381 


3769 
6670 


4000 
6959 


4351 
7248 


4641 
7536 


4932 

7825 


5222 
8113 


5512 
8401 


6802 
8689 


291 
239 


176091 


151 


8977 


9264 


9552 


9839 


.126 


.413 


.699 


.985 


1272 


1558 


287 


152 


181S44 


2129 


2415 


2700 


2985 


3270| 3555 


3839 


4123 


4407 


286 


153 


4691 


4975 


5259 


5542 


5825 


6108 


6391 


6674 


6956 


7239 


283 


154 


7521 


7S03 


80S4 


8366 


8647 


8928 


9209 


9490 


9771 


..51 


281 


155 


190332 


0612 


0892 


1171 


1451 


1730 


2010 


2289 


2667 


2846 


279 


156 


3125 


3403 


3681 


3959 


4237 


4514 


4792 


6069 


5346 


6623 


378 


157 


5899 


6176 


6453 


6729 


7005 


7281 


7556 


7832 


8107 


8382 


276 


158 


8657 


8932 


9206 


9481 


9755 


..29 


.303 


.577 


.850 


1124 


274 


159 ' 201397 


1670' 19431 2216 


2488 


27611 30331 33051 35771 3848 


272 


N. 1 |l|2|3 4|5|6|7|8|9|D. 1 



^? 



raan 1 to lO.OOO. 



|a|3i4|5|»l7;al9|P . 



I1120i 43911 4fi&l'-UIM 
68!G 709e, 7365 7631 
)>5isla7a3 ..SI .319 
81313813451 S7a) 29S6 
tsa 5109' S373 563^ 
74^*4 7717 eOlO si73 
;il|(l^ 0370 0631: <)-!.4-> 
■;7lfi,2976 3236 319G 



5^H: btli' !1716' 
7901,3173 tlUl' 
.53S! .«tS3 USI 
325S 3318 37^ 

5903 eisfi 6iau 
dt9S 9UW) 
IIU 1675 
lOld. 4S71' 
6Seu 6^59 
9170 9430 



3r5S 



T 8913 



■::lj& 3350. 3501 3767 
fi.i^iS 5781' 0033 Bias 
^l)4l•l 8297:8513 8799 
1' 49 0799 1048' 1897 
t Hi] 3336 aT34 378" 
Ula7a9 6006 6352 

i 8 19 8484 

Ilff4 090S 

J9B 
398 



ilb , 0" 


01 






.yiuii'i38i 


11- 


3W1 3^ 


.1 










2256 1 3' 


44158 4fiS7 





17^4 1979' 
am 1517. 
6789 7041 < 
9399 9550, 
17951 SOU' 
1. 15S5 
674S 6991 1 
9193 0448 

ea^ iim\ 

4061 «806 | 
61 7 6718 
t.S77 91 lA' 
1^63, IWll 
Tfi.Wl'-W 3' 
3990,0 U, 
sm 8^78, 
6/Jt 91Sl 
3UJ1 J.JJI 
Sill 5^1. 
7ri)0 78381 



6016 6306 8556<I71 
8710 8979 9347: 369 
13^ 1651' I93|l 367 
1019 4311 1579: ~ " 
6<19t 6957 T33I 
9333 95(<5 9846: _ , 
1936 3196 3156:361 
1.133 479S 50.<ilj3S9 

7115 S37S i«w'a " 

9688 8933 .IM lS 
SS34.S188WIWS 

4770 sosaunfa 

7393 75U 7796 353 
9800 ..90| ..lOoU"' 
2393 fiSll''- n, ■ 
4773 5019 r I 

"" 748 , 



3214 



9687 093 

2610 
4548 4i90 5031 
6958 TIUI 7439 

OBM OSH 

1-39 1979 
1109 434S 4d- 
We? 670J ha 

i^u 80IS a 

1141 137 I H 

aib'liaeyi) lu 

577i 6003,1) . 
8067 S39bI 85 



— i 


htl 


■*U 


»II4I> 


li 


'UK'. 


4fiM 


W1'' 


H)f 


am 


(.Til 


117 W 


Hiid 


Hi.i 


T-lfiM 




H7S5 


H'lm 


HI '11 


1348 


mn 


lift 11 




IX'lf 


1477 


^rM 




mm 


3Mfi 


3447 


IfiSf 




ftiiM 


fil'.m 


M!VH 


ftm 


n8f;[ 




7KHI 


Tisn 


S65H 


SH5Ji 90S1 


Hvs: 


M51 


0143 




H):i!) 


U37 


1435 



liJI 


1 


■^ ' 








BIB 6731 


.1 II 


Hni 


^fil 7781 


71I7J 


N1711 


HfO 9801 






1630 183!l 






3610 38BC 


4051 




5068 nSSE 


(irisii 


OVflll 


76.'i0 7BBE 


H'lftti 


Hll.'iT 


flnSO, U345 


47 




n-iai IR30 


3J)3H 


ssaf. 



I e I 7 I » I ■ TT 





4 


A TABLE OF LOGARITHMS FROM 1 TO 10,000. 








N. 


1 |l[2|3 4|5 6 7|8|9|D. 1 




220 


342423, 2620 


2817 


3014 3212, 3409, 3606, 3802 


3999 4196 


197 




221 


4392 


4589 


4786 


4981 


6178 


6374 


6570 


6766 


5962 


6167 


196 




222 


6353 


6549 


6744 


6939 


7135 


7330 


7625 


7720 


7915 


8110 


195 




223 


8305 


8500 


8694 


8889 


9083 


9278 


9472 


9666 


9860 


..54 


194 




224 


350248 


0442 


0636 


0829 


1023 


1216 


1410 


1603 


1796 


1989 


193 




225 


2183 


2375 


2568 


2761 


2954 


3147 


3339 


3632 


3724 


3916 


193 




226 


4108 


4301 


4493 


4685 


4876 


6068 


6260 


6462 


6643 


6834 


192 




227 


6026 


6217 


6408 


6599 


6790 


6981 


7172 


7363 


7564 


7744 


191 




228 


7935 


8125 


8316 


8606 


8696 


8886 


9076 


9266 


9466 


9646 


190 




229 
23a 


9835 


..25 
1917 


.216 
2105 


.404 
2294 


.593 
2482 


.783 
2671 


.972 
2859 


1161 
3048 


1350 
3236 


1639 
3424 


189 
188 


1 


361728 


1 


231 


3612 


3800 


3988 


4176 


4363 


4561 


4739 


4926 


5113 


5301 


188 




232 


5488 


6675 


6862 


6049 


6236 


6423 


6610 


6796 


6983 


7109 


187 




233 


7356 


7542 


7729 


7916 


8101 


8287 


8473 


8669 


8846 


9030 


186 




234 


9216 


9401 


9587 


9772 


9968 


.143 


.328 


.618 


.698 


.883 


185 




235 


371068 


1253 


1437 


1622 


1806 


1991 


2176 


2360 


2544 


2728 


184 




236 


2912 


3096 


3280 


3464 


3047 


3831 


4015 


4198 


4382 


4565 


184 




237 


4748 


4932 


5115 


5298 


6481 


5664 


6846 


6029 


6212 


6394 


183 




238 


6577 


6759 


6942 


7124 


7306 


7488 


7670 


7862 


8034 


8210 


182 




239 
240 


8398 


8580 
0392 


8761 
0573 


8943 
0754 


9124 
0934 


9306 
1115 


9487 
1296 


9668 
1476 


9849 
1656 


..30 
1837 


181 
181 




880211 




241 


2017 


2197 


2377 


2557 


2737 


2917 


3097 


3277 


3456 


3836 


180 




242 


3815 


3995 


4174 


4353 


4533 


4712 


4891 


5070 


6249 


5428 


179 




243 


5806 


5785 


5964 


6142 


0321 


6499 


6677 


6856 


7034 


72121 178 




244 


7390 


7568 


7746 


7923- 


8101 


8279 


8456 


8634 


8811 


8989! 178 




245 


9166 


9343 


9520 


9693 


9875 


..51 


.228 


.405 


.582 


.759 177 




246 


390935 


1112 


12S8 


1404 


1641 


1817 


1993 


2169 


2345 


2.521 


176 




247 


2697 


2373 


3048 


3224 


3100 


3575 


3751 


3926 


4101 


427i^ 


176 




248 


4452 


4827 


4802 


4977 


5152 


5326 


5501 


5676 


5850 


6025 


176 




249 


6199 


6374 


6548 


0722 


6896 


7071 


7245 


7419 


7592 


7766 174 1 




250 


337940 


8114 


8237 


84«1 


8834 


8303 


8931 


91.54 


9328 


9501 


173 




251 


9674 


9847 


..20 


.192 


.365 


.538 


.711 


.883 


10.56 


1228 


173 




252 


401401 


1573 


1745 


1917 2089 


2261 


2433 2605 


2777 


2949 


172 




253 


3121 


3292 


3464 


3635 


3807 


3978 


4149 


4320 


4492 


4663 


171 




254 


4834 


5005 


5176 


5346 


55.7 


5638 


5358 


6029 


6199 


6370 


171 




255 


6511) 


6710 


6881 


7051 


7221 


7391 


7561 


7731 


7901 


8070 


170 




256 


8210 


8410 


8579 


8749 


S'JIS 


9037 


9257 


9426 9595 


9764 


169 




257 


9933 


.102 


.271 


.440 .609 


.777 


.916 


1114 


1283 


1451 


169 




258 


411620 


1783 


1956 


2124 2293 


2461 


2629 


2790 


2964 


3132 


168 




259 


3300 


3467 


3635 


3303 3970 


4J37 


4305 


4472 


4639 


4808 


167 




260 


414973 


5140 


5307 


6474 5G41 


5308 


5974 


6141 


6303 


6474 


167 




261 


6641 


6807 


6973 


7139 7306 


7472 


7633 


7304 


7970 


8135 


166 




282 


8301 


8467 


8633 


8793 


8964 


9129 


9295 


9-160 


9825 


9791 


165 




263 


9956 


.121 


.286 


.451 


.616 


.781 


.945 


1110 


1275 


1439 


165 




264 


421604 


17CS 


1933 


20;)7 


2261 


2128 2590 


2754 


2918 


3082 


164 




265 


3246 


3410 


3574 


3737 


3901 


4065 


4223 


^392 


4555 


4718 1641 




266 


4882 


5045 


5208 


.5371 


5534 


5697 


5360 


6023 


6186 


6349 


163 


267 


0511 


mi'ir 


6S36 


6999 


7161 


7324 


7488 


7648 


7811 


7973 


162 


268 


8135 


8297 


8459 


8621 


8783 


8944 


9106 


9268 


9429 


9591 


162 




269 
270 


9752 


9914 
1.525 


..75 
1685 


.236 
1846 


.398 
2007 


.559 
2167 


.720 
2328 


.881 
2438 


1042 
5>349 


1203 
2809 


161 
161 




431364 




271 


2969 


3130 


3290 


3450 3610 


3770 


3930 


4090 ! 4249 


4409 


160 




272 


4569 


4729 


4888 


5048 


5207 


.5367 


6526 


5685 5844 


6004 


169 


! 

1 


273 


6163 


6322 


6481 


6640 


6798 


6957 


7116 


7275 7433 


7592 


159 


1 


274 


7751 


7909 


8067 


8226 


8394 


8542 8701 


885919017 


9175 


158 




275 


9333 


9491 


9648 


9806 


9964 


.122 .279 


.437 


.594 


.7.52 


158 




276 


440909 


1066 


1224 


1.381 


1538 1695; 1852 


2009 


2166 


2323 


157 




277 


2480 


9637 


2793 


2950 3l06j 3263 3419 3576,3732 


3889 


157 




278 


4045 


4201 


4357 


4513 4669 4825' 4931 5 1 37 j 5293 


5449 


166 




279 


5604 


5760' 5915 


6071 6226 6382' 6537 6692' 6848 


7003 1551 




N. 


10 ll2|3|4 5 6|7 8|9!D. 1 



« 





A TABLS OF LOOARITHllS FKOX 1 


TO 10,000. 


t 


6 


N. 


0Fl|2 3t4|6 6|7!8 9|D. | 


280 


447158 


7^18 


7468 


7623 


7778 


7933 


8088 


8242 


8397 8552i 155 1 


281 


8706 


8861 


9015 


9170 


9324 


9478 


9633 


9787 


9941 


..95 


154 


282 


450249 


0403 


0657 


0711 


0865 


1018 


1172 


1326 


1479 


1633 


154 


283 


1786 


1940 


2093 


2247 


2400 


2553 


2706 


2859 


3012 


3165 


153 


284 


3318 


3471 


3624 


3777 


3930 


4082 


4235 


4387 


4540 


4692 


153 


285 


4845 


4997 


6150 


5302 


5454 


5606 


5758 


5910 


6062 


6214 


162 


286 


6366 


6518 


6670 


6821 


6973 


7125 


7276 


7428 


7579 


7731 


152 1 
151 1 
151 1 


287 


7882 


8033 


8184 


8336 


8487 


8638 


8789 


8940 


9091 


9242 


288 


9392 


9543 


9694 


9845 


9995 


.146 


.296 


.447 


.597 


..748 


289 


46089S 


1048 


1198 


1348 


1499 


1649 


1799 


1948 


2098 


2248 


150 


290 


462398 


2548 


2697 


2847 


2997 


3146 


3296 


3445 


3594 


3744 


150 


291 


3893 


4042 


4191 


4340 


4490 


4639 


4788 


4936 


5085 


5234 


149 


292 


5383 


5532 


5680 


5829 


5977 


6126 


6274 


6423 


6571 


6719 


149 


293 


6868 


7016 


7164 


7312 


7460 


7608 


7756 


7904 


8052 


8200 


148 


294 


8347 


8495 


8643 


8790 


8938 


90S5 


9233 


9380 


9627 


9675 


148 


295 


9822 


9969 


.116 


.263 


.410 


.56# 


.704 


.851 


.998 


1145 


147 


296 


471292 


1438 


1585 


1732 


1878 


2025 


2171 


2318 


2464 


2610 


146 


297 


2756 


2903 


3049 


3195 


3341 


3487 


3633 


3779 


3925 


4071 


146 


298 


4216 


4362 


4508 


4653 


4799 4944 


5090 


5235 


5381 


5526 


146 


299 


5671 


5816 


5962 


6107 


6252 6397 


6542 


6687 


6832 


6976 


145 


300 


477121 


726(i'7.111 


7655 


7700i 7844- 7<»^9' 81:^3 


8278 


8422 


145 


301 


8566 


8711 


8855 


899!^ 


9143 


928719431 UOVS 


9719 


9863 


144 


302 


480007 


0151 


0294 


0438 


0582 


0725 0869 


1012 


1156 


1299 


144 


303 


1443 


1586 


1729 


1872 


2016 


2159 2302 


2445 


2688 


2731 


143 


304 


2874 


3016 


3159 


3302 


3445 


3587 


3730 


3872 


4015 


4157 143 1 


305 


4300 


4442 


4585 


4727 


48G9 


6011 


5153 


5295 


5437 


5579 :42| 


306 


5721 


58G3 


6005 


6147 


6289 


6430 


6572 


6714 


6855 


G997 


142 


307 


7138 


7280 


7421 


7563 


7704 


7845 


7986 


8127 


8269 


8410 


141 


308 


8551 


8692 


8833 


8974 


9114 


9256 


9396 


9537 


9677 


9818 


141 


309 
310 


9958 


..99 
1502 


1642 


.380 
1782 


.520 
1922 


.661 
2062 


.801 
2201 


.941 
2341 


1081 
2481 


1222 
2621 


140 
140 


491362 


311 


2760 


2900 


3040 


3179 


3319 


3458 


3597 


3737 


38V6 


4015 


139 


312 


4155 


4294 


4433 


4572 


4711 


4850 


4989 


5128 


5267 


5406 


139 


313 


5644 


5683 


5822 


5960 


6099 


6238 


6376 


6515 


6653 


6791 


139 


314 


6930 


7068 


7206 


7344 


7483 


7621 


7759 


7897 


8035 


8173 


138 


315 


8311 


8448 


8586 


8724 


8862 


8999 


9137 


9275 


9412 


9550 


138 


316 


9687 


9824 


9962 


..99 


.236 


.3'i4 


.511 


.648 


.785 


.922 


137 


317 


501059 


1196 


1333 


1470 


1607 


1744 


1880 2017 


2154 


j^.aoi 


137 


318 


2427 


25H4 


2700 


2837 


2973 


3109 


3246! 3382 


3518 


3655 


136 


319 
3*^0 


3791 


3927 
5286 


4063 
5421 


4199 
.5557 


4335 
5693 


4471 

5828 


4607 


4743 


4878 


5014 


136 
136 


505150 


5964! C099 


6234 6370 


321 


6505 


6640 


6776 


6911 


7046 


7181 


7316 


7451 


7586 7721 


135 


322 


7856 


7991 


8126 


8260 


8395 


8530 


8664 


8799 


8934 


9068 


135 


323 


9203 


9337 


9471 


9606 


9740 


9874 


. . . »f 


.143 


.277 


.411 


134 


824 


510545 


0679 


0813 


0947 


1081 


1215 


1349 


1482 


1616 


1750 


134 


325 


1883 


2017 


2151 


2284 


2418 


2551 


2684 


2818 


2951 


3084 


133 


826 


3218 


3351 


3484 


3617 


3750 


3883 


4016 


4149 


4282 


4414 


133 


827 


4548 


4681 


4813 


4946 


5079 


6211 


5344 


6476 


5609 


5741 


133 


828 


5874 


6006 


6139 


6271 


6403 


6535 


6668 


6800 


6932 


7064 


132 


329 
330 


7196 


7328 
8^46 


7460 

8777 


7592 
8909 


7724 
9040 


7856 
9171 


7987 
9303 


8119 
9434 


8251 
9566 


8382 
9697 


132 
131 


518514 


331 


9628 


9959 


..90 


.221 


.353 


.484 


.616 


.745 


.876 


1007 


131 


332 


521138 


1269 


1400 


1530 


1661 


1792 


1922 


2053 


2183 


2314 


131 


333 


2444 


2575 


2705 


2835 


2966 


3096 


3226 


3356 


3486 


3616 


130 


884 


3746 


3876 


4006 


4136 


4266 


4396 


4526 


4656 


4785 


4915 


130 


835 


5045 


5174 


5304 


5434 


5563 


5693 


5822 


5951 


6081 


6210 


129 


886 


6339 


6469 


6598 


6727 


6856 


6986 


7114 


7243 


7372 


7501 


129 


337 


7630, 7759 

8917 9046 

530200 0328 


7888 


8016 


8145 


8274 


8402 


8531 


8660 


8788 


129 


8:m 


9174 


9302 


9430 


9559 


9687 


9815 


0943 


..72 


128 


839 

N. 


0456 


0584 


0712' 08401 0968< 1096 


1223 


1351 


128 


ll 2|3|4|5|6|7|8|9|D. 1 



6 


A TABLE OF LOGARITHMS FUOM 1 


TO 10,000 






N. 


|l|2|3 4|5|6|7|8|0|D. 1 


340 


531479 


1607 


1734 


1862 


1990 


2117 


2245 


2372 


2500 


i:627 


128 


341 


2754 


2882 


3009 


3136 


3264 


3391 


3518 


3645 


3772 


3899 


127 


342 


4026 


4153 


4280 


4407 


4534 


4661 


4787 


4914 


5041 


5167 


127 


343 


5294 


5421 


5547 


5674 


6800 


5927 


6053 


6180 


6306 


6432 


126 


344 


6558 


6685 


6811 


6937 


7063 


7189 


7315 


7441 


7567 


7693 


126 


345 


7819 


7945 


8071 


8197 


8322 


8448 


8574 


8699 


8826 


8961 


126 


346 


9076 


9202 


9327 


9462 


9578 


9703 


9829 


9964 


..79 


.204 


125 


347 


540329 


0455 


0680 


0705 


0830 


0955 


1080 


1205 


1330 


14.54 


125 


348 


1579 


1704 


1829 


1963 


2078 


2203 


2327 


2452 


2576 


2701 


125 


349 
350 


2825 


2950 
4192 


3074 
4316 


3199 
4440 


3323 
4564 


3447 

4688 


3571 
4812 


3696 
4936 


38ro 

5060 


3944 

6183 


124 
124 


544068 


351 


5307 


5431 


6555 


6678 


5802 


6925 


6049 


6172 


6296 


6419 


124 


352 


6543 


6666 


6789 


6913 


7036 


7169 


7282 


7405 


7529 


7652 


123 


353 


7775 


7898 


8021 


8144 


8267 


8389 


8512 


8635 


8758 


8881 


123 


354 


9003 


9126 


9249 


9371 


9494 


9616 


9739 


9861 


9984 


.106 


123 


355 


550228 


0351 


0473 


0596 


0717 


0840 


0962 


1084 


1206 


1328 


122 


356 


1450 


1572 


1694 


1816 


1938 


2060 


2181 


2303 


2425 


2547 


122 


367 


2668 


2790 


2911 


3033 


3166 


3276 


3398 


3519 


3640 


3762 


121 


358 


3883 


4004 


4126 


4247 


4368 


4489 


4610 


4731 


4852 


4973 


121 


359 
360 


5094 


5215 
6423 


5336 
6544 


5457 
6664 


5578 
6785 


5699 
6906 


5820 
7026 


6940 
7146 


6061 
7267 


6182 
7387 


121 
120 


556303 


361 


7507 


7627 


7748 


7868 


7988 


8108 


8228 


8349 


8469 


8589 


120 


362 


8709 


8829 


8948 


9068 


9188 


9308 


9428 


9548 


9667 


9787 


120 


363 


9907 


..26 


.146 


.265 


.385 


.504 


.624 


.743 


.863 


.982 


119 


364 


561101 


1221 


1340 


1459 


1578 


1698 


1817 


1936 


2056 


2174 


119 


365 


2293 


2412 


2531 


2650 


2769 


2887 


3006 


3125 


3244 


3362 


119 


366 


3481 


3600 


3718 


3837 


3955 


4074 


4192 


4311 


4429 


4548 


119 


367 


4666 


4784 


4903 


5021 


6139 


5257 


5376 


5494 


6612 


5730 


118 


368 


5848 


6966 


6084 


6202 


6320 


6437 


0555 


6673 


6791 


6909 


118 


369 
370 


7026 


7144 
8319 


7262 
8436 


7379 
8664 


7497 
8671 


7614 

8788 


7732 

8906 


7849 
9023 


7967 
9140 


8084 
9257 


118 
117 


568202 


371 


9374 


9491 


9608 


9725 


9842 


9959 


..76 


.193 .309 


.426 


117 


372 


670543 


0660 


0776 


0893 


1010 


1126 


1243 


1359 


1476 


1592 


117 


373 


1709 


1825 


1942 


2058 


2174 


2291 


2407 


2523 


2639 


2756 


116 


374 


2872 


2988 


3104 


3220 


3336 


3452 


3668 


3684 


3800 


3915 


IW 


375 


4031 


4147 


4263 


4379 


4494 


4610 


4726 


4841 


4957 


5072 


116 


376 


5188 


5303 


5419 


5534 


5650 


5765 


6880 


5990 


6111 


6326 


115 


377 


6341 


6457 


6572 


6687 


6802 


6917 


7032 


7147 


7262 


7377 


115 


378 


7492 


7607 


7722 


7836 


7951 


8066 


8181 


8295 


8410 


8525 


115 


379 
380 


8639 


8754 
9898 


8868 
..12 


8983 
.126 


9097 
.241 


9212 
.355 


9326 
.469 


9441 
.583 


9555 
.697 


9669 
.811 


114 
114 


579784 


381 


580925 


1039 


1163 


1267 


1381 


1495 


1608 


1722 


1836 


1950 


114 


382 


2063 


2177 


2291 


2404 


2518 


2631 


2746 


2858 


2972 


3085 


114 


383 


3199 


3312 


3426 


3639 


3652 


3765 


3879 


3992 


4105 


4218 


113 


384 


4331 


4444 


4557 


4670 


4783 


4896 


6009 


5122 


5235 


5348 


113 


385 


5461 


5674 


5686 


5799 


5912 


6024 


61.37 


6250 


6362 


6475 


113 


386 


6587 


6700 


6812 


6926 


7037 


7149 


7262 


7374 


7486 


7599 


lis 


387 


7711 


7823 


7936 


8047 


8160 


8272 


8384 


8496 


8608 


8720 


112 


388 


8832 


8944 


9066 


9167 


9279 


9391 


9603 


9615 


9726 


9838 


112 


389 
390 


9950 


..61 
1176 


.173 

1287 


.284 
1399 


.396 
1510 


.607 
1021 


.619 
1732 


.730 
1843 


.842 
1955 


.953 
2066 


112 
111 


591065 


391 


2177 


2288 


2399 


2510 


2621 


2732 


2843 


2954 


3064 


3175 


lU 


392 


3286 


3397 


3508 


3618 


3729 


3840 


3950 


4061 


4171 


4282 


111 


393 


4393 


4503 


4614 


4724 


4834 


4945 


6055 


5166 


5276 


5386 


110 


394 


6496 


5606 


5717 


5827 


6937 


6047 


6167 


626* 


6377 


6487 


110 


395 


6597 


6707 


6817 


6927 


7037 


7146 


7256 


7366 


7476 


7586 


110 


396 


7695 


7805 


7914 


8024 


8134 


8243 


8353 


8462 


8572 


8681 


lU) 


397 


8791 


8900 


9009 


9119 


9228 


9337 


9446 


9556 


9665, 9774 


109 


398 


9883 9992 


.101 


.210 


.319 


.428 


.537 


.646 


.756 


.864 


109 


399 


600973 1082 


1191 


1299 


1408 


1617 


1625 


1734 


1843 


l\fb\ 


109 


N. 


|l|2|3|4|5|6 7l8l9in. 1 





A TART.B OF LOGABITHIIS FBOai 1 TO 10,000. 




7 


F 


Oil 2 3 4|5 6|7!8|9|D| 


1400 


602060 


2169 


2277 


2386, 2494| 2603] 


2711 


2819 


2928 


3036 


108 


401 


3144 


3253 


3361 


3469 


3577 


3686 


3794 


3902 


4010 


4118 


106 


402 


4226 


4334 


4442 


4550 


4658 


4766 


4874 


4982 


5089 


5197 


108 


403 


5305 


5413 


5621 


5628 


5736 


5844 


5951 


6059 


6166 


6274 


108 


404 


6381 


6489 


6696 


6704 


6811 


6919 


7026 


7133 


7241 


7348 


107 


405 


7456 


7562 


7669 


7777 


7884 


7991 


8098 


8205 8312 


8419 


107 


406 


8526 


8633 


8740 


8847 


8954 


906 1 


9167 


9274 


9381 


9488 


107 


407 


95941 9701 


9808 


9914 


..21 


.128 


.234 


.341 


.447 


.554 


107 


408 


610660 


0767 


0873 


0979 


1086 


1192 


1298 


1405 


1511 


1617 


106 


409 
410 


1723 


1829 
2890 


1936 
2996 


2042 
3102 


2148 
3207 


2254 
3313 


2360 
3419 


2466 
3525 


2572 
3630 


2678 
3736 


106 
106 


612784 


411 


3842 


3947 


4053 


4159 


4264 


4370 


4475 


4581 


4686 


4792 


106 


412 


4897 


5003 


5108 


5213 


5319 


5424 


5529 


5634 


5740 


5845 


105 


413 


5950 


6055 


6160 


6265 


6370 


6476 


6581 


6686 


6790 


6895 


105 


414 


7000 


7105 


7210 


7315 


7420 


7526 


7629 


7734 


7839 


7943 


105 


415 


8048 


8153 


8257 


8362 


8466 


8671 


8676 


8780 


8884 


8989 


105 


416 


9093 


9198 


9302 


9406 


9611 


9616 


9719 


9824 


9928 


..32 


104 


417 


620136 


0240 


0344 


0448 


0552 


0656 


0760 


0864 


0968 


1072 


104 


418 


1176 


1280 


1384 


1488 


1592 


1695 


1799 


1903 


2007 


2110 


104 


419 
420 


2214 


2318 
3363 


2421 
3456 


2525 
3559 


2628 
3663 


2732 
3766 


2835 
3869 


2939 
3973 


3042 
4076 


3146 
4179 


104 
103 


623249 


421 


4282 


4386 


4488 


4591 


4695 


4798 


4901 


5004 


5107 


5210 


103 


422 


5312 


5415 


5618 


5621 


5724 


5827 


5929 


6032 


61361 6238 


103 


423 


6340 


6443 


6546 


6648 


6751 


6853 


6966 


7058 


7161 


7263 


103 


424 


7356 


7468 


7571 


7673 


7775 


7878 


7980 


8082 


8185 


8287 


102 


425 


8389 


8491 


8693 


8606 


8797 


8900 


9002 9104 


9206 


9308 


102 


426 


9410 


9512 


9613 


9715 


9817 


9919 


..211 .123 


.224 


.326 


102 


i27 


630428 


0530 


0631 


0733 


0835 


0936 


1038 


1139 


1241 


1342 


102 


428 


1444 


1545 


1647 


1748 


1849 


1951 


2052 


2153 


2255! 2366 


101 


429 
430 


2457 


2559 
3569 


2660 
3670 


2761 
3771 


2862 
3872 


2963 
3973 


3064 
4074 


3165 
4175 


3266 
4276 


3367 
4376 


101 
100 


633468 


431 


4477 


4578 


4679 


4779 


4880 


4981 


5081 


5182 


6283 


5383 


100 


432 


5484 


6584 


5686 


5786 


5886 


6986 


6087 


6187 


6287 


6388 


100 


433 


6488 


6588 


6688 


6789 


6889 


6989 


7089 


7189 


7290 


7390 


100 


434 


7490 


7590 


7690 


7790 


7890 


7990 


8090 


8190 


8290 


8389 


99 


435 


8489 


8589 


8^89 


8789 


8888 


8988 


90881 9188 


9287 


9387 


99 


436 


9486 


9586 


9686 


9785 


9885 


9984 


..84 


.183 


.283 


.382 


99 


437 


640481 


0581 


0680 


0779 


0879 


0978 


1077 


1177 


1276 


1375 


99 


438 


1474 


1573 


1672 


1771 


1871 


1970 


2069 


2168 


2267 


2366 


99 


439 
440 


2465 


2503 
3551 


2662 
3650 


2761 
3749 


2860 
3347 


2969 
3946 


3058 
4044 


3156 
4143 


3265 

4242 


3354 
4340 


99 
98 


643453 


441 


4439 


4537 


4036 


4734 


4832 


4931 


5029 


5127 


5226 


5324 


98 


442 


5422 


5521 


5619 


5717 


5815 


5913 


6011 


6110 


6208 


6306 


98 


443 


6404 


0502 


6600 


6698 


6796 


6894 


6992 


7089 


7187 


7285 


98 


444 


7383 


7481 


7579 


7676 


7774 


7872 


7969 


8067 


8165 


8262 


98 


445 


8360 


8458 


8655 


8663 


8760 


8848 


8945 


9043 


9140 


9237 


97 


446 


9335 


9432 


9530 


9627 


9724 


9821 


9919 


..16 


.113 


.210 


97 


447 


650308 


0405 


0502 


0599 


0696 


0793 


0890 


0987 


1084 


1181 


97 


44S 


1278 


1375 


1472 


1569 


1666 


1762 


1869 


1956 


2053 


2150 


97 


1449 


2246 


2343 
3309 


2440 
3405 


2536 
3502 


2633 
3698 


2730 
3696 


2826 
3791 


2923 
3888 


3019 
3984 


3116 
4080 


97 
96 


450 


653213 


451 


4177 


4273 


4369 


4465 


4562 


4658 


4754 


4850 


4946 


5042 


96 


452 


5138 


5235 


5331 


5427 


5523 


5619 


5715 


5810 


5906 


6002 


96 


453 


6098 


6194 


6290 


6386 


6482 


6577 


6673 


67691 6864 


6960 


96 


454 


7050 


7162 


7247 


7343 


7438 


7534 


'7629 


7725 


7820 


7916 


96 


455 


8011 


8107 


8202 


8298 


8393 


8488 


8684 


8679 


8774 


8870 


95 


456 


8965 


9060 


9155 


9250 


9346 


9441 


9536 


9631 


9726 


9821 


95 


457 


9916 


..11 


.106 


.201 


.296 


.391 


.486 


.581 


.676 


.771 


95 


458 


660865 


0960 


1055 


1150 


1245 


1339 


1434 


1529 


16231 1718 


95 


459 


1813 


1907 


2002 


2096 


2191 


228G 


2380 


2475 


256912653! 95 | 


N. 


0ll|2i3|4|5|6|7|8 9|D. | 








• 




Cc* 















8 


A TABLB OF L06ABITHX8 FBOH 1 


TO 10,000. 


• 




N. 


|0il|2 3|4|6|6|7|8 9|D. | 


460 


662758 


2852 2947 


3041 3135|3230|3324|3418 


3512 3607, 941 


461 


3701 


3795 


3889 


3983 


4078 


4172 


4266 


4360 


4454 


4548 


M 


462 


4642 


4736 


4830 


4924 


6018 


5112 


6200 


5299 


5393 


6487 


94 


463 


5581 


6675 


5769 


5862 


69.56 


6060 


6143 


6237 


6331 


6424 


94 


464 


6518 


6612 


6706 


6799 


6892 


6986 


7079 


7173 


7266 


7360 


94 


165 


7453 


7546 


7640 


7733 


7826 


^7920 


8013 


8106 


8199 


8293 


93 


466 


8386 


8479 


8572 


8665 


8769 


8852 


8946 


9038 


9131 


9224 


93 


467 


9317 


9410 


9503 


9596 


9689 


9782 


9876 


9967 


..60 


.153 


93 


468 


670246 


0339 


0431 


0524 


0617 


0710 


0802 


0896 


0988 


1080 


93 


469 
470 


1173 


1265 
2190 


1368 
2283 


1451 
2375 


1643 
2467 


1636 
2560 


1728 
2662 


1821 
2744 


1913 
2836 


2005 
2929 


93 
92 


672098 


471 


:^021 


3113 


3205 


3297 


3390 


3482 


c:74 


3666 


3758 


38,50 


92 


472 


3942 


4034 


4126 


4218 


4310 


4402 


4494 


4586 


4677 


4709 


92 


473 


4861 


4953 


5046 


5137 


5228 


5320 


6412 


6603 


5596 


5687 


92 


474 


5778 


5870 


6962 


6053 


6145 


6236 


6328 


6419 


6511 


6602 


92 


475 


6694 


6785 


6876 


6968 


7059 


7151 


7242 


7333 


7424 


7516 


91 


476 


7607 


7698 


7789 


7881 


7972 


8063 


8164 


8245 


8336 


8427 


91 


477 


8518 


8609 


8700 


8791 


8882 


8973 


9064 


9155 


9246 


9337 


91 


478 


9428 


9519 


9610 


9700 


9791 


9882 


9973 


..63 


.164 


.246 


91 


479 


680336 


0426 


0617 


0607 


0698 


0789 


0879 


0970 


1060 


1161 


91 


480 


681241 


1332 


1422 


1513 


1603 


1693 


1784 


1874 


1964 


2066 


90 


481 


2146 


2235 


2326 


2416 


2506 


2696 


2686 


2777 


2867 


2967 


90 


482 


3047 


3137 


3227 


3317 


3407 


3497 


3587 


3677 


3767 


3857 


90 


483 


3947 


4037 


4127 


4217 


4307 


4396 


4486 


4576 


4666 


4756 


90 


484 


4845 


4935 


6025 


5114 


5204 


6294 


6383 


5473 


6663 6652 


90 


485 


5742 


5831 


5921 


6010 


6100 


6189 


6279 


6368 


6458 


6647 


89 


486 


6636 


6726 


6815 


6904 


6994 


7083 


7172 


7261 


7361 


7440 


89 


487 


7529 


7618 


7707 


7796 


7886 


7976 


8064 


8153 


8242 


8331 


89 


488 


8420 


8509 


8598 


8687 


8776 


8865 


8953 


9042 


9131 


9220 


89 


489 


9309 


9398 


9486 


9675 


9664 


9763 


9841 


9930 


..19 


.107 


89 


490 


690196 


0285 


0373 


0462 


0550 


0639 


0728 


0816 


0906 


0993 


89 


491 


1081 


1170 


1258 


1347 


1435 


1524 


1612 


1700 


1789 


1877 


88 


492 


1965 


2053 


2142 


2230 


2318 


2406 


2494 


2583 


2671 


2759 


89 


493 


2847 


2935 


3023 


3111 


3199 


3287 


3375 


3463 


3651 


3639 


88 


494 


3727 


3815 


3903 


3991 


4078 


4166 


42.')4 


4342 


4430 


4517 


88 


495 


4605 


4693 


4781 


4868 


4956 6044 


6131 


6219 


5307 


5394 


88 


496 


6482 


5569 


5657 


5744 


5832 5919 


6007 


6094 


6182 


6269 


87 


497 


6356 


6444 


G531 


6618 


6706 1 6793 


6880 


6968 705517142 


87 


498 


7229 


7317 


7404 


7491 


7.578:7665 


7752 


7839 


7926 


8014 


87 


499 


8101 


8188 


8275 


8362| 84491 8.535 


8622 


8709 


8796 


8883 


87 


500 


698970 


9057 


9144 


9231 


9317; 9404 


9491 


9578 


9664 


9751 


87 


501 


9838 


9924 


..11 


..98 


.1841 .271 


.3.58 


.444 


.631 


.617 


87 


502 


700704 


0790 


0877 


0963 


10.50 1136 


1222 


1.309 


1395 


1482 


86 


503 


1568 


1654 


1741 


1827 


1913 


1999 


2080 


2172 


2258 


2344 


86 


504 


2431 


2517 


2603 


2689 


2776 


2861 


2947 


3033 


3119 


3205 


86 


505 


3291 


3377 


3463 


3549 


3635 


3721 


3807 


3895 


3979 


4065 


86 


506 


4151 


4236 


4322 


4408 


4494 


4679 


4665 


4751 


4837 


4922 


86 


507 


6008 


5094 


5179 


5266 


6350 


6436 


5522 


5607 


6693 


5778 


86 


508 


5864 


5949 


6035 


6120 


6206 


6291 


6376 


6462 


6547 


6632 


85 


509 
510 


6718 


G803 
7655 


6888 
7740 


6974 
7826 


7059 
7911 


7144 
7996 


7229 
8081 


7315 
8166 


7400 
8251 


7485 
8336 


85 
85 


707570 


511 


8421 


8506 


8591 


8676 


8761 


8846 


8931 


9015 


9100 


9185 


66 


612 


9270 


9365 


9440 


9624 9609 


9694 


9779 


9863 


9948 


..33 


85 


513 


710117 


0202 


0287 


0371 


0456 


0.540 


0625 


0710 


0794 


0879 


85 


514 


0963 


1048 


1132 


1217 


1301 


1385 


1470 


15.54 


1639 


1723 


84 


515 


1807 


1892 


1976 


20G0 


2144 


2229 


2313 


2397 


2481 


2566 


84 


516 


2650 


2734 


2818 


2902 


2986 


3070 


3154 


3238 


3323 


3407 


84 


617 


3491 


3575 


3650 


3742 


3826 


3910 


3994 


4078 


4162 


4246 


84 


518 


4330 


4414 


4497 


4.581 


4665 


4749 


4833 


4916 


5000 


5084 


84 


519 


5167^ 52511 5335' 54181 5602' 5580 


5669 


5753 5836 


5920 


84 


N. 


oil 2|3 4} 5 6|7l8!9|D. 1 



A TABLE OF LOOAXITBHS TKOTt 1 TO 10,000. 



N. 1 1 I 1 a 1 3 1 4 1 5 1 R 1 7 1 8 I fl 


D. 




6!0 


716003,6087,6170 


6354 


6337, 6431 


^504 




^w' 




G21 




693 


7004 


7088 


7171 


7354 


7338 


7421 7504 




83 






767 


7754 


7837 


7930 


80O; 


8086 


8169 


8253 8336 








633 


850: 


8585 


8668 


8751 




8917 


9000 


9083 9165 








fi34 






















S3 




G25 




















090: 


83 




5SG 












646 




S3 




637 


1 






3469 




83 




638 


1 


291 




82 




fiS9 








S2 




m 








63 




531 










83 




633 






504 




83 




633 












37E 


7461 


8 




634 




33 








48 


s 












535 


M 


843 










84 




OOi 


9084 






639 
637 




346 






94 






40 


813 
631 


9893 
.702 






638 


30 




H 










34 


438 


1608 






633 
MO 


1589 


1669 
474 


1750 
"665 


1830 
36i5 


1911 


1991 
3796 


072 
2876 


0,50 

^956 


323^ 
3037 


"413 


|- 




tTIH 




541 


aioT 


3 78 


^3 8 


n-ia 


3518| 3598 


3679 


3759 


3S39 








543 




40 J 110) -I 1(J 


41 o' IKK 




4W0 


47 






543 
544 


481)- 
5 It 


, , , ,,« 


80 
80 




S45 


b3J7 nt 


80 




US 


711) m 


79 




647 
648 


9H 1 01 
8 ■^l 1 J J l-t J49} 


79 

79 




549 


9 bt 


47 


12h 


205 


384 


79 




650 


710 1 )757 


0436 


0bT6 


0994 


1073 


79 




551 


11 1 46 


16 4 


1703 


17S2 




79 




ass 


191 


1| )33 


^411 


489 


^s' 


364B 


79 




553 






31 et 


^ 79 


3431 


78 




654 


3 10 






1 1 T ■) 






405S 


4136 


4 15 






666 


4 J^ 




4449 




4000 








1414 


1J97 


78 




660 


50 


515t 












5H31 


6699 


5777 


78 




667 


5856 












C323 


0.101 


6479 


B5 6 


78 




568 


D<i34 














71 9 


7356 


334 


78 




669 
















7 95 J 




8110 


78 




660 








sial 










8808 


55^6 


77 




661 








9U-. 






942 


9604 


9 82 


9659 






563 








9968 




123 




377 


354 








663 






0663 


0740 




09)11 U I I01« 


11 5 


120i 






564 








1510 


l-SS 1 1 1 1 197" 






6S5 








279 


3 r 40 






566 




89J 29 


301 


31 i 1 OE 










3f (10 3716 


3Sn 












4-1 1 01 


1 7 ir I 036 


76 




5S0 




1 1 6749 


76 




57? 


WJ( 168 


MHi 6860 
"441 7320 


"76 
76 








8003, 8079 


76 








87611 8836 


76 




674 




9517i 0508 


76 




876 


9668 J n 4 IJ J J4 1 7I> 4 121 196 


372 .347 


76 




676 


7604 S 04as 0573 00491 0734 0799 0875 0950 


1035 110 


76 




677 


1176 1 51 1336 1403 147 1553 1037 1703 


17781 1853 


76 




678 


1938 001 0/8SI5Stj 8 33012378 3453 


25S9. 3604 


76 




679 


2679 -7S4 .S.9l3U04'^,8 306313138 3303 1 3378 ! 3353' 76 




H. 1 l> 1 1 1 3 1 3 1 4 1 3 1 6 1 7 I 8 1 i- ■ - 





10 



A TABLE OF LOGABmmS FAOJC 1 TO 10,000. 



N. I 



Iir2|3i4|5|6|7|8|9|0. 



680 

581 
682 
583 
584 
585 
586 
587 
588 
589 

590 
591 
592 
593 
594 
595 
596 
597 
598 
599 

600 
601 
602 
603 
604 
605 
606 
607 
608 
609 

610 
611 
612 
613 
614 
615 
616 
617 
618 
619 

620 
621 
622 
623 
624 
625 
626 
627 
628 
629 

630 
631 
632 
633 
634 
635 
636 
637 
638 
639 



763428 
4176 
4923 
5669 
6413 
7156 
7898 
8638 
9377 

770115 



770852 
1587 
2322 
3055 
3786 
4517 
5246 
5974 
6701 
7427 



778151 
88*/4 
9596 

780317 
1037 
1755 
2473 
3189 
8904 
4617 



785330 
6041 
6751 
7460 
8168 
8876 
9581 

790285 
0988 
1691 



3503 
4251 
4998 
5743 
6487 
7230 
7972 
8712 
9451 
0189 



0926 
1661 
2395 
3128 
3860 
4590 
5319 
6047 
6774 
7499 



8224 
8947 
9669 
0389 
1109 
1827 
2544 
3260 
3975 
4689 



792392 
3092 
3790 

44881 
51851 
6880; 
6574; 
7208i 
7960! 
8651 



799341 
800029 
0717 
1404 
2089 
2774 
3467 
4139 
4821 
5501 



6401 
6112 
6822 
7631 
8239 
8946 
9651 
0356 
1059 
1761 

2462 
3162 
3860 
4558 
5254 
5949 
6644 
7337 
8029 
8720 



3578 
4326 
5072 
5818 
6562 
7304 
8046 
8786 
9526 
0263 



0999 
1734 
2468 
3201 
8933 
4663 
6392 
6120 
6846 
7572 



8296 
9019 
9741 
0461 
1181 
1899 
2616 
3332 
4046 
4760 



8653 
4400 
6147 
6892 
6636 
7379 
8120 
8860 
9599 
0336 



1073 
1808 
2642 
3274 
4006 
4736 
6465 
6193 
6919 
7644 



5472 
6183 
6893 
7602 
8310 
9016 
9722 
0426 
1129 
1831 



2532 
3231 
3930 
4C27 
5324 
6019 
6713 
7406 
8098 
8789 



8368 
9091 
9813 
0533 
1253 
197] 
2688 
3403 
4118 
4831 



3727 
4475 
6221 
5966 
6710 
7453 
8194 
8934 
9673 
0410 

1146 
1881 
2615 
3348 
4079 
4809 
5538 
6265 
6992 
7717 



6543 
6254 
6964 
7673 
8381 
9087 
9792 
0496 
1199 
1901 



8441 
9163 
9885 
0605 
1324 
2042 
2759 
3475 
4189 
4902 



3802 
4650 
6296 
0041 
6786 
7627 
8268 
9008 
9746 
0484 



1220 
1955 
2688 
3421 
4162 
4882 
5610 
6338 
7064 
7789 



6616 
6325 
7035 
7744 
8451 
9157 
9863 
0567 
1269 
1971 



9409 

i0098 

0786 

1472 
2158 
2842 
3525 
4208 
4889 
5569 



9478 
0167 
0854 
1541 
2226 
2910 
3594 
4276 
4957 
5G37 



8513 
9236 
9957 
0677 
1396 
2114 
2831 
3546 
4261 
4974 



2602 
3301 
4000 
4697 
5393 
6088 
6782 
7475 
8167 
8868 

9547! 

0236 

0923 

16091 

2295 

2979 

3662 

4344 

5025 

5705 



2672 
3371 
4070; 

4767. 
5463 



6158 



6852! 

75451 
8236; 
8927 



9610 

0305; 

09921 

1678: 

2363' 

3047; 

3730 

4412,1 

5093 

5773' 



5686 
6396 
7106 
7815 
8622 
9228 
9933 
0637 
1340 
2041 

2742 
3441 
4139 
4S3G 
5532 
6227 
6921 
7614 
8305 
8996 

9685 

0373 

1061 

1747 

2432 

3116: 

3798' 

4480i 

5161' 

5841' 



3877 
4624 
6370 
6116 
6859 
7601 
8342 
9082 
9820 
0567 



1293 
202s 
2762 
3494 
4225 
4955 
5683 
6411 
7137 
7862 



3952 
4699 
6446 
6190 
6933 
7675 
8416 
9156 
9894 
0631 



8585 
9308 
..29 
0749 
1468 
2186 
2902 
3618 
4332 
5045 



5757 
6467 
7177 
7885 
8593 
9299 
...4 
0707 
1410 
2111 



1367 
2102 
2835 
3567 
4298 
6028 
5756 
6483 
7209 
7934 



8658 
9380 
.101 
0821 
1540 
2258 
2974 
3689 
4403 
511 6 

6828 
6538 
7248 
7956 
8663 
9369 
..74 
0778 
1480 
2181 



2812; 

35I1I 

4209! 

4906! 

5002: 

G-2\i7 

6990| 

7683; 

8374! 

9065 



2882 
3581 
4279 
4976 
5672 
6366 
7060 
7752 
8443 
9134 



4027 
4774 
5620 
6264 
7007 
7749 
8490 
9230 
9968 
0706 



1440 
21 r5 
2908 
3640 
4371 
6100 
5829 
6556 
7282 
8006 



8730 
9452 
.173 
0893 
1612 
2329 
3046 
3761 
4476 
5187 



4101 
4848 
5594 
6338 
7082 
7823 
8564 
9303 
..42 
0778 



1514 
2248 
2981 
3713 
4444 
5173 
5902 
6629 
7354 
8079 



5899 
6609 
7319 
8027 
8734 
9440 
.144 
0848 
1550 
2252 



8802 
9524 
.245 
0965 
1684 
2401 
3117 
3832 
4546 
5259 



5970 
6680 
7390 
8098 
8804 
9510 
.215 
0918 
1620 
2322 



9754 
0442 
1129 
1815 
2500 
31841 
3867! 
4548i 
52291 
59O81 



9823 
0511 
1198 
1884 
2568 
3252 
3935 
4616 
52'J7 
5!)"'6 



2952 
3651 
4349 
.5045 
5741 
6436 
7129 
7821 
8513 
9203 



9892 
0580 
1266 
1952 
2637 
3321 
4003 
4685 
5365 
6044 



3022 
3721 
4418 
5115 
5811 
6505 
7198 
7890 
8582 
9272 



9961 
0648 
1335 
2021 
2705 
3389 
4071 
4753 
.'343.) 
6112 



76 
T6 
75 
74 
74 
74 
74 
74 
74 
74 

74 
78 
78 
78 
78 
78 
73 
73 
73 
7% 

72 
72 
72 
72 
72 
72 
72 
71 
71 
71 



71 
71 
71 
71 
71 
71 
70 
70 
70 

70 
70 
70 
70 
70 
69 
69 
69 
69 
69 

69 
69 
69 
69 
69 
68 
68 
68 
68 
68 



3as: 



N. I I « I 2 I 3 I 4 



5 I 6 t 7 i 8 I 9 I D. 



. A ^r«^H0^k7Ml^ 



J 



A TABLE OF LOOARrrtOtS FROM T TO 10,000. 



11 



JK. 


1. 


1 1 


1 « 


1 8 


1 * 


1 5 


1 6 


1 7 


1 8 


1 9 


111. 


640 


806180 


6248 


6316 


6384 


6451 


6519 


65871 


6655| 6723 


07901 681 


641 


6858 


6926 


6994 


7061 


7129 


7197 


7264 


7332 


7400 


7407 


68 


642 


7535 


7603 


7670 


7738 


7806 


7873 


7941 


8008 


8076 


8143 


68 


643 


8211 


8279 


8346 


8414 


8481 


85-19 


8616 


8684 


8751 


8818 


67 


644 


8886 


8953 


9021 


9088 


9156 


9223 


9290 


9358 


9425 


9492 


67 


645 


9560 


9627 


9694 


9762 


9829 


9896 


9964 


..31 


..98 


.165 


67 


646 


810233 


0300 


0367 


0434 


0501 


0569 


0636 


0703 


0770 


0837 


67 


647 


0904 


0971 


1039 


1106 


1173 


1240 


1307 


1374 


1441 


1508 


67 


648 


1575 


1642 


1709 


1776 


1843 


1910 


1977 


2044 


2111 


2178 


67 


619 
650 


2245 


2315 
2980 


2379 
3047 


2445 
3114 


2512 
3181 


2579 
3247 


2646 
3314 


2713 
3381 


2780 

3448 


2847 
3514 


6/ 
67 


812913 


651 


8581 


3648 


3714 


3781 


3848 


3914 


3981 


4048 


4114 


4181 


67 


652 


4248 


4314 


4381 


4447 


4514 


4581 


4647 


4714 


4780 


4847 


67 


653 


4913 


4980 


50-16 


5113 


5179 


5246 


.5312 


5378 


5445 


6511 


66 


654 


5578 


5644 


5711 


5777 


5843 


5910 


5976 


6042 


6109 


6175 


66 


655 


6241 


6308 


6374 


6440 


6506 


6573 


6639 


6705 


6771 


6838 


66 


656 


6904 


6970 


7036 


7102 


7169 


7235 


7301 


7367 


7433 


7499 


66 


657 


7565 


7631 


7698 


7764 


7830 


7896 


7962 


8028 


8094 


8160 


66 


658 


8226 


8292 


8358 


8424 


8490 


8556 


8622 


8688 


8764 


8820 


66 


659 
660 


8885 


8951 
9610 


9017 
9676 


9083 
9741 


9149 
9807 


9215 
9873 


9281 
9939 


9346 

...4 


9412 
..70 


9478 
.136 


66 
66 


819544 


661 


820201 


0267 


0333 


0399 


0464 


0530 


0595 


0661 


0727 


0'"92 


66 


662 


0858 


0924 


0989 


1055 


1120 


1186 


1251 


1317 


1382 


1448 


66 


663 


1514 


1579 


1645 


1710 


1775 


1841 


1906 


1972 


2037 


2103 


65 


6G4 


2 1 68 


2233 


2299 


2364 


2430 


2495 


2560 


2626 


2691 


2756 


65 


665 


2822 


2887 


2952 


3018 


3083 


3148 


3213 


3279 


3344 


3409 


65 


606 


3474 


3539 


3605 


3670 


3735 


3800 


3865 


3930 


3996 


4001 


65 


667 


4126 


4191 


4256 


4321 


4386 


4451 


4516 


4581 


4646 


4711 


65 


668 


4776 


4841 


4906 


4971 


5036 


5101 


5166 


5231 


5296 


5301 


65 


669 


5426 


5491 


5556 


5621 


5686 


5751 


5815 


5880 


5945 


0010 


66 


670 


826075 


6140 


6204 


6269 


6334 


6399 


6464 


6528 


0593 


0058 


65 


671 


6723 


6787 


6852 


6917 


6981 


7046 


7111 


7175 


7240 


7306 


65 


672 


7369 


7434 


7499 


7563 


7628 


7692 


7757 


7821 


7880 


7951 


66 


673 


8015 


8080 


8144 


8209 


8273 


8338 


8402 


8467 


8531 


8595 


64 


674 


8660 


8724 


8789 


8853 


8918 


8982 


9046 


9111 


9175 


9239 


64 


675 


9304 


9368 


9432 


9497 


9561 


9625 


9690 


9754 


9818 


9882 


64 


676 


9947 


;.ll 


..75 


.139 


.204 


.268 


.332 


•396 


.400 


.525 


64 


677 


830589 


0653 


0717 


0781 


0845 


0909 


0973 


1037 


1102 


1100 


64 


678 


1230 


1294 


1358 


1422 


1486 


15.50 


1614 


1078 


1742 


1800 


64 


679 


1870 


1934 


1998 


2062 


2126 


2189 


2253 


2317 


2,381 


2445 


64 


680 


832509 


2573 


2637 


2700 


2764 


2828 


2892 


2956 


3020 


3083 


64 


681 


3147 


3211 


3275 


3338 


3402 


3466 


3530 


3593 


3057 


3721 


64 


682 


3784 


3848 


3912 


3075 


4030 


4103 


4166 


4230 


4294 


4357 


64 


683 


4421 


4484 


4648 


4611 


4675 


4739 


4802 


4800 


4929 


4993 


64 


684 


5056 


5!20 


5183 


5247 


;;310 


5373 


,5437 


5500 


5504 


5027 


63 


685 


5691 


5754 


5817 


5881 


5944 


6007 


6071 


6134 


0197 


0261 


63 


686 


6324 


6387 


6451 


6514 


0577 


6641 


6704 


6707 


0830 


0894 


63 


687 


6957 


7020 


7083 


7146 


7210 


7273 


7336 


7399 


7402 


7525 


63 


688 


7588 


7652 


7715 


7778 


7841 


7904 


7907 


8030 


8093 


8150 


63 


689 


8219 


8282 


8345 


8408 


8471 


§534 


8597 


8660 


8723 


8780 


63 


690 


838849 


8912 


8975 


9038 


9101 


9164 


9227 


9289 


9352 


9415 


63 


691 


9478 


9541 


9604 


9667 


9729 


9792 


9855 


9918 


9981 


..43 


63 


692 


840106 


0169 


0232 


0294 


0357 


0420 


0482 


0545 


0008 


0071 


63 


693 


0733 


0796 


0859 


0921 


0984 


1046 


1109 


1172 


1234 


1297 


63 


694 


1359 


1422 


1485 


1547 


1610 


1672 


1735 


1797 


1800 


1922 


63 


695 


1985 


2047 


2110 


2172 


2235 


2297 


2360 


2422 2484 


2547 


62 


696 


2609 


2672 


2734 


2796 


2859 


2921 


2983 


3046 


3108 


3170 


62 


697 


3233 


3295 


3357 


3420 


3482 


3544 


3606 


3669 


3731 


3793 


62 


608 


3855 


3918 


3980 


4042 


4104 


4166 


4229 


4291 


4353 


4415 


62 


699 


4477 


4539 


4601 


4604 


4726 47881 


48,50 


4912 4974J5036! 62 | 


N. 


0|l|2 3i4 6|6 7|8|9|D. i 



16 



12 



A TABLE OF LOOABITHMS FROM 1 TO 10,000. 



N. 


1 1|2|3|4|5|6|7|8|9|D. 


700 


845098 


6160> 5222 


6284 


6346i 6408 


6470 


6632 


6594 


6656 


63 


701 


6718 


6780. 6842 


6904 


6966 


6028 


6090 


6151 


6213 


6275 


62 


702 


6337 


6399 


6461 


6523 


6586 


6646 


6708 


6770 


6832 


6894 


62 


705 


6955 


7017 


7079 


7141 


7202 


7264 


7326 


7388 


7449 


7611 


62 


704 


7573 


7634 


7696 


7768 


7819 


7881 


7943 


8004 


8066 


8128 


62 


705 


8189 


8251 


8312 


8374 


8435 


8497 


8559 


8620 


8682 


8743 


62 


706 


8805 


8866 


8928 


8989 


9051 


9112 


9174 


9236 


9297 


9358 


61 


707 


9419 


9481 


9642 


9604 


9665 


9726 


9788 


9849 


9911 


9972 


61 


708 


850033 


0096 


0156 


0217 


0279 


0340 


0401 


0462 


0524 


0585 


61 


709 
710 


0646 


0707 
1320 


0769 
1381 


0830 
1442 


0891 
1503 


0952 
1564 


1014 
1625 


1076 
1686 


1136 
1747 


1197 


61 
61 


851258 


1809 


711 


1870 


1931 


1992 


2053 


2114 


2175 


2236 


2297 


2358 


2419 


61 


ri2 


2480 


2641 


2602 


2663 


2724 


2785 


2846 


2907 


2968 


3029 


61 


713 


3090 


3160 


3211 


3272 


3333 


3394 


3466 


3616 


3577 


3637 


61 


714 


3698 


3769 


3820 


3881 


3941 


4002 


4003 


4124 


4185 


4245 


61 


715 


4306 


4367 


4428 


4488 


4649 


4610 


4670 


4731 


4792 


4852 


61 


716 


4913 


4974 


6034 


6095 


6156 


5216 


6277 


6337 


6398 


6459 


61 


717 


5519 


6680 


6640 


5701 


6761 


6822 


6882 


6943 


6003 


6064 


61 


718 


6124 


6186 


,6246 


6306 


6366 


6427 


6487 


6648 


6608 


6668 


60 


719 
720 


6729 


6789 
7393 


6850 
7453 


6910 
7513 


6970 
7574 


7031 
7634 


7091 
7694 


7162 
7765 


7212 
7816 


7272 

7875 


60 
60 


857332 


721 


7936 


7995 


8056 


8116 


8176 


8236 


8297 


8367 


8417 


8477 


60 


722 


8537 


8597 


8657 


8718 


8778 


8833 


8898 


8968 


9018 


9078 


60 


723 


9138 


9198 


9258 


9318 


9379 


9439 


9499 


9559 


9619 


9679 


60 


724 


9739 


9799 


9859 


9918 


9978 


..38 


..98 


.168 


.218 


.278 


60 


725 


860338 


0398 


0468 


0518 


0578 


0637 


0697 


0757 


0817 


0877 


60 


726 


0937 


0996 


1056 


1116 


1176 


1236 


1296 


1356 


1415 


1476 


60 


727 


1534 


1594 


1664 


1714 


1773 


1833 


1893 


1952 


2012 


2072 


60 


728 


2131 


2191 


2261 


2310 


2370 


2430 


2489 


2549 


2608 


2668 


60 


729 
730 


2728 


2787 
3382 


2847 
3442 


2906 
3501 


2966 
3561 


3025 
3620 


3085 
3680 


3144 
3739 


3204 
3799 


3263 
3868 


60 


863323 


69 
69 


731 


3917 


3977 


4036 


4096 


4165 


4214 


4274 


4333 


4392 


4452 


732 


4511 


4570 


4630 


4689 


4748 


4803 


4867 


4926 


4986 


6045 


69 


733 


5104 


5163 


6222 


5282 


5341 


5400 


6459 


6519 


6578 


6637 


69 


734 


6696 


6755 


6814 


6874 


5933 


5992 


6051 


6110 


6169 


6228 


69 


735 


6287 


6346 


6405 


6465 


6524 


6583 


6842 


6701 


6760 


6819 


69 


73b 


6878 


6937 


6996 


7055 


7114 


7173 


7232 


7291 


7350 


7409 


69 


737 


7467 


7526 


7685 


7644 


7703 


7762 


7821 


7880 


7939 


7998 


59 


738 


8056 


8115 


8174 


8233 


8292 


8350 


8409 


8468 


8527 


8586 


59 


739 
740 


8644 


8703 
9290 


8762 
9349 


8821 
9408 


8879 
9466 


89.38 
95*^6 


8997 
9584 


9050 
9642 


9114 
9701 


9173 
9760 


59 
69 


869232 


741 


9818 


9877 


9935 


9994 


..53 


.111 


.170 


.228 


.287 


.345 


69 


742 


870404 


0462 


0521 


0579 


or>38 


0696" 


U755 


0813 


0872 


0930 


68 


743 


0989 


1047 


1106 


1164 


1223 


1281 


1339 


1398 


1456 


1615 


58 


744 


1573 


1631 


1690 


1748 


1806 


•1865 


1923 


1981 


2040 


2098 


68 


745 


2156 


2215 


2273 


2331 


2389 


2448 


2506 


2564 


2622 


2681 


58 


746 


2739 


2797 


2855 


2913 


2972 


3030 


3088 


3146 


3204 


3262 


58 


747 


3321 


3379 


3437 


3495 


3553 


3611 


3669 


3727 


3785 


3844 


58 


7AS 


3302 


3960 


4018 


4076 


4134 


4192 


4250 


4308 


4366 


4424 


58 


749 
750 


4482 


4540 
5119 


4598 
5177 


4656 
5235 


4714 
5293 


4772 
5351 


4830 
5409 


4888 
5466 


4946 
5524 


6003 
6682 


58 
58 


875061 


751 


5640 


5698 


6756 


5813 


5871 


5929 


5987 


6045 


6102 


6160 


58 


752 


6218 


6276 


6333 


6391 


6449 


6507 


6564 


6622 


6630 


6737 


58 


753 


6795 


6353 


6910 


6968 


7026 


7083 


7141 


7199 


7256 


7314 


58 


754 


7371 


7429 


7487 


7544 


7602 


7659 


7717 


7774 


7832 


7889 


58 


765 


7947 


8004 


8062 


8119 


8177 


82C4 


8292 


8349 


8407 


8464 


57 


756 


8522 


8579 


8637 


8694 


8752 


8809 


8366 


8924 


8931 


9039 


57 


757 


9096 


9153 


9211 


9268 


9325 


9383 


9140 


9497 


9556 


9612 


57 


"758 


9669 


9726 


9784 


9341 


9898 


9956 


..13 


,.70 


.127 


.185 


57 


759 


880242 


0299 


0366 


041310471 


0528 


0585 


j642 


0699 


0766 


57 


_N. 


|l|2|n 4|5|6|7|8|9|D. 





A 


TIXL 


s OF LooABiraKs ntoM 1 


It>l( 


,000 




13 


s. 


1 1 1 3 1 3 1 4 1 5 1 6 1 7 1 8 1 9 1 D. 1 


7S5- 


8S0314 0871, VU-iS. 09851 i(M3, 1099, 1156, 1-J13; Vifl- 1328, 57 1 


761 


1385 


1442 


1499 


1556 1013 


1670 


1727 


1784 




1)496 


£7 


763 


1955 


2012 


2069 


2126 


3183 


2240 


2297 


2354 




2468 


61 


763 


2525 


26B1 


aeas 


3695 


3752 


2809 


2866 


3933 


3980 


3037 


57 


704 


3093 


3150 


mm 


3384 


8381 


3377 


3434 


8491 


3648 


3606 


57 


765 


3661 


3718 


3776 


3839 


38S8 


3»46 


4D02 


4068 


4115 


4172 


57 


766 


4229 


4385 


434S 


439B 


4456 


4613 


4G69 


4635 


4682 


4730 


57 


767 


4795 


485S 


4wg 


49BS 


5038 


6078 


6135 


6193 


6348 


6305 


57 


76a 


5361 


5418 


6474 


5531 


5587 


6644 


5700 


5757 


5613 


6870 


57 


769 
770 


5920 


5983 
6547 


6039 
6604 


0096 
6660 


6151 

67lfl 


6209 


8365 
68^ 


6331 




6434 
6998 


56 

66 


8B6«iT 


771 


7054 


7111 


7167 




7260 


7336 


7392 






7561 


66 


772 


76(7 


7G74 


7730 




7642 


789fi 


7956 


8011 


6067 


8123 


66 


773 




8336 


8393 


8348 


«i404 


6460 


8516 


8573 


8629 




56 


774 


8741 


8797 


8353 


8309 


6Qaa 


9031 


9077 


9134 


9190 


9246 


56 


77B 


9302 


935S 


9414 


9470 


5638 


a-iSs 


9638 


9684 


9760 


9808 


56 


77fl 


9S62 


9918 


9974 


30 


86 


141 






.309 




66 


777 


390421 


0477 


0-513 


0589 0641 


0700 


OTSS 


0812 


0666 




56 


778 


01BO 




lOUi 


1147 1 0^ 


1259 


1314 


1370 


1436 




56 


779 




181 S 


1872 


IB28 


1983 


2039 


56 


780 






2429 


a4S4 


3640 


3005 


-58 


TSl 


Sf.l 2929 


298S 


3040 


8096 


3151 


56 


7H3 


a I 3484 


S-ilO 


3595 


3651 


3706 


56 


783 


a7U- 1 


4031 


4004 


4150 


4306 


4301 


55 


784 




4j71 








4593 


4648 


4704 


4759 


4814 


55 


785 




4925 




5036 


5091 


5146 


5201 


5367 


5313 


6367 


SI 


786 




5478 




6588 


5644 


S699 




5809 


6864 


6930 


55 


787 




«030 




6140 




62-.1 


6308 


6361 


6416 


6471 


55 


783 




658 1 


6036 


6692 


6747 


6609 


6867 


69 3 


6967 


7033 


5 


789 




7132 

7683 


~ 


7MS 

7792 


7297 
7947 


7903 


7107 
7957 


74 3 

80 2 


7517 

8067 


7572 
8i23 


56 


897627 




8176 


8231 




8341 


8396 


8451 


8506 


85 1 


8015 


8670 


55 


782 


8736 




8835 




8944 


8999 


9064 


91 9 


9164 


9318 


5S 




9273 


9328 


9383 


»437 


9492 


9517 


9603 


96 6 


9711 


9766 


55 


794 


0831 


0875 


9930 


9965 


..39 




.149 


.3 3 


.368 


.313 


56 


795 


900367 


0423 


0470 


4531 


0586 


0640 


0695 


0749 


0804 


0850 


55 




0913 


0968 


1033 


1077 


1131 


1186 


1240 


1295 




1404 


6 




14S8 


1513 


1667 


1833 


1676 


mi 




1840 


1894 


1918 


54 




2003 


B057 


Slli 


■2166 


2281 






2384 




8492 


64 


800 


3517 


360 
3144 


3T9E 


3710 
3253 


3764 
3;!07 


3361 


3476 


3470 


2981 
3624 


3038 
3578 


6 
6 


9a3O90 


801 




368 


3741 


3795 


3849 


3904 


3958 


4013 


4066 


4120 


64 


SOS 


4174 


432 


4283 


4337 


439J 


4445 


4499 


4553 


4607 


4661 


a 


803 


4716 


4770 


4824 


487H 




49.S0 


5040 


5094 


514* 


5203 


804 


6356 


53 


5304 


54 a 


547i 


£626 


63B0 


5634 




5742 


6 


806 


6796 


5850 


6901 


5 S 


6012 


6066 


6118 


61-/3 


6221 


0281 


54 


806 


6335 


63 


«443 


6497 


6561 


6604 


6658 


6713 


6766 


0820 


54 


807 


6374 


69 


6981 


7 5 


708£ 


7143 


7198 


7260 


7304 


7358 


54 


80« 


7411 


7465 


7519 


7 3 




7680 


im 


7781 


7841 


7895 


fi 


nog 


7949 


8003 


8066 


S 


81fl! 


8317 






8378 


8431 


5 


m 


908485 


8539 


8592 


8 4 


8699 


8763 


8801 


sm 


8914 


8987 


-M 


81 


9021 


9074 


9128 


9 a 


9235 


923 


9343 


B396 


9449 


9503 


64 


Bia 


055b 


9610 


9663 


9 8 


9770 


983 


9877 


993( 


9964 


..37 


6 


s a 


910091 


0144 


0197 


026 


0304 


0358 


041 


0464 


0516 


0571 


5 


a 4 


0624 


0678 


0731 


4 


0838 


089 


0944 


0998 


1U5I 


1104 


5 


a a 


1168 


Hi 


1284 


1 7 


1371 


1424 


1477 


1530 


lfi84 


1B37 


5 


a 


1690 


1743 


179- 


1 


1903 


1956 


3009 


3063 


2116 


2169 


5 


a 


8223 


3376 


3328 


£ 


3435 


248 


364 


2594 


3647 


270O 


5 


81 


3753 


3800 


3869 


3 


2966 


301 


3072 


3126 


3178 


329) 


5 


31 


3-284 


33371 3390 


344 


3496 


354 


3602 


3056 


3709 


376 


6S 


N. 


i 1 1 2 1 3 1 4 1 5 1 6 1 7 1 8 1 





14 



▲ TABLE OF L06ASITHM8 FBOM I TO 10)000. 



N. 


l!2 3 4|5 6|7|8|9 D. | 


820 


913814,8867 


3920 


3973 4026 


4079i 4132i 4184 


4237 


42901 531 


821 


4343 4396 


4U9 


4502 


4665 


4608 


4660 


4713 


4766 


4819 


53 


822 


4872 4925 


4977 


6030 


.5083 


6136 


5189 


6241 


6294 


6347 


68 


823 


5400 5453 


5505 


6558 


5611 


6664 


5716 


6769 


6822 


6875 


53 


824 


5927 


5980 6033 


6085 


6138 


6191 


6243 


6296 


6349 


6401 


53 


825 


6454 


6507 


6659 


6612 


6664 


6717 


6770 


6822 


6876 


6927 


53 


826 


69801 7033 


7085 


7138 


7190 


7243 


7295 


7348 


7400 


7463 


63 


827 


7506 i 7558 


7611 


7663 


7716 


7768 


7820 


7873 


7926 


7978 


62 


828 


8030*8083 


8135 


8188 


8240 


8293 


8346 


8397 


8460 


8602 


53 


829 


8555 1 8607 


8669 


8712 


8764 


8816 


8869 


8921 


8973 


9026 


53 


830 


919078; 9130 


9183 


9235 


9287 


9340 


9392 


9444 


9496 


9649 


53 


831 


9601 9653 9706 


9758 


9810 


9862 


9914 


9967 


..19 


..71 


52 


832 


9201231 0176 


0228 


0280 


0332 


0384 


0436 


0489 


0541 


0593 


62 


833 


0645 


0697 


0749 


0801 


0863 


0906 


09.58 


1010 


1062 


1114 


62 


834 


1166 


1218 


1270 


1322 


1374 


1426 


1478 


1630 


1582 


1634 


62 


835 


1686 1738 


1790 


1842 


1894 


1946 


1998 


2060 


2102 


21.54 


62 


836 


2206 


2258 


2310 


2362 


2414 


2466 


2518 


2670 


2622 


2674 


62 


837 


2725 


2777 


2829 


2881 


2933 


2986 


3037 


3089 


3140 


3192 


53 


838 


3244 


3296 


3348 


3399 


3461 


3603 


3555 


3607 


3668 


3710 


52 


839 
840 


3762 


3814 
4331 


as65 

4383 


3917 
4434 


3969 
4486 


4021 
4.538 


4072 
4589 


4124 
4641 


4176 
4693 


4228 


53 
52 


924279 


4744 


841 


4796 


4848 


4899 


4951 


6003 


60.54 


5106 


61.57 


6209 


6261 


52 


842 


6312 


6364 


5416 


5467 


6518 


5570 


.5621 


5673 


6725 


6776 


53 


843 


6828 


6879 


5931 


5982 


6034 


6085 


6137 


6188 


6240 


6291 


51 


844 


6342 


6394 


6445 


6497 


6548 


6600 


6651 


6702 


6764 


6805 


51 


845 


6857 


6908 


6959 


7011 


7062 


7114 


7165 


7216 


7268 


7319 


51 


846 


7370 


7422 


7473 


7,524 


7576 


7627 


7678 


7730 


7781 


7832 


51 


847 


7883 


7935 


7986 


8037 


8088 


8140 


8191 


8242 


8293 


8346 


51 


848 


83961 8447 


8498 


8549 


8601 


8662 


8703 


8764 


8806 


8867 


51 


849 
850 


8908 


8959 
9470 


9010 
9521 


9061 
9572 


9112 
9623 


9163 9215 


9266 
9776 


9317 
9827 


9368 


51 
61 


929419 


9674 


9725 


9879 


851 


9930 


9981 


..32 


..83 


.134 


.185 


.236 


.287 


.338 


.389, 611 


852 


930440 


0491 


0.542 


0592 


0643 


0694 


0745 


0796 


0847 


0898 


51 


853 


0949 


1000 


1051 


1102 


11.53 


1204 


12.54 


i:i05 


1366 


1407 


51 


854 


1458 


1509 


1.560 


1610 


1661 


1712 


1763 


1814 


1866 


1915 


51 


865 


1966 


2017 


2068 


2118 


2169 


2220 2271 


2322 


2372 


2423 


51 


856 


2474 


2524 


2575 


2626 


2677 


2727 2778 


2829 


2879 


2930 


51 


857 


2981 


3031 


3082 


3133 


3183 


3234 3285 


3335 


3386 


3437 


51 


858 


3487; 3538 


3589 


3639 


3G90 


3740:3791 


3841 


3892 


3943 


61 


859 
860 


3993 


4044 
4549 


4094 
4599 


4145 
4650 


4195 
4700 


4246 4296 


4347 
4852 


4397 


4448 
4953 


61 
60 


934498 


4751 


4801 


4902 


861 


5003 1 5054 


5104 


5154 


5205 


.5255 


5306 


5356 


5406 


.5457 


60 


862 


5507 5558 


5608 


5658 5709 


5759: .5809' 5860 


.5910 


5960 


60 


863 


6011 6061 


6111 


6162 6212 


6262' 6313 0363 


6413 


6463 


60 


864 


6514! 6564 6614 


666516715 


6765; 6815^ 0865 


6916 


6966 


60 


8C5 


7016; 7066: 7117 


716717217 


7267! 7317 


7367 


7418 


7468 


60 


866 


7518; 7568 7618 


7668 7718 


7769 


7819 


7869 


7919 


7**69 


60 


807 


8019,8069 8119 


8169 8219 


8269i 8320 


8370 


8420 


8470 


60 


SOS 


8520, 8570; 8620 


8670. 8720' 87701 8820! 8870 


8920 


8970 


60 


869 


9020: 9070 


9120 


9170 9220 


9270 9320 9369 


9419 


9469 


60 


870 


939519 9569 


9610 


966 J 9719 


9769!98iyi 9869 


9918 


9968 


60 


871 


940018 0008 


0118 


0108; 0218; 0267! 0317' 0367 


0417 


0467 


.50 


872 


0516 0r)66l 0610 


0666 


07160765 0815, 0865 


0915 


09641 60 1 


873 


1014! 1064 


1114 


1163 1213i 1263^ 1313; 1362 


1412 1462 60 1 


874 


1511' 1561 


1611 


1660! 1710: 1760 1809| 1859 


1909 1958 60 


876 


2008 2058 


2107 


2157 


2207i2258 23061 2355, 2405! 2455! 50 1 


876 


2501; 2554, 2603 


2053 


2702 


2752 2801 2851 


2901 2950; 60 


877 


3000; 3049 


i 3099 


3148 


3198 


32471 3297, 3346 


3396 34451 49 


878 


34951 3544 


i 3593 


.36431 3692! 3712 3791 


3841 


38901 3939 49 


879 


a089l 4038' 4088' 41371 41861 4236' 42S51 4335 


43S4'4433 49 


N. 


! 1 2 1 3 1 4 1 5 • 6 1 7 1 8 ' 9 


n 





A TASLE tlF LOOAXTIUMa FBUM 1 TO 10 


000. 




lb 


^ 


1 1 1 3 1 3 1 4 1 5 1 6 1 7 1 8 1 1 D. 1 


B80 


944433 4533 






4630 1 47^9 






4aJ7|49S7 49 


881 


4976 502r> 








6223 


5373 


5a;i 


6370 5419 49 


882 


5464 






5616 


5665 


5715 


5764 


6813 


6SG3 5912 




883 


5961 


tiOlO 


B050 


6108 


6157 


6307 


6356 


6305 


6364 


6403 


49 


esi 


6453 


G501 


6551 


6600 


6649 




6747 


6796 


6846 


6894 


49 


SS5 


6943 


6932 


7041 


7090 


7140 


7189 


7339 


7287 


7336 


7385 


49 


SSS 


7434 


7483 


7532 


7S81 


7630 


7679 


7739 


7777 


7S26 


7875 


49 


8ST 






8022 


8070 


8119 


9168 


8317 


S266 


8315 


8364 


40 


SBS 


8413 


8462 


8511 


8560 


8609 


8667 


8706 


8755 


8804 


8853 


49 


889 

890 


S902 


895! 


8999 


9048 
9^ 


9097 


9146 
9634 


9196 


9244 

97JI 


9292 
9780 


9341 
9^ 


49 
49 


81H90 




89 1 


9878 


99 6 99 5 










319 


267 


316 


49 


893 


950365 04 llnil 




1(0 MS 




06 


0754 


08(3 


49 




03f= ' 1 1 1 119 


1240 


1389 


49 




13J ! l^^^ 


1726 


177 




845 


18 3 1 lt-3 


221 1 


60 


48 


8SS 


230 1 17 


696 


741 


48 


897 




MSO 


3 8 




898 


3 1 t i d7ll 




899 


376.1 M 1194 


48 


BOO 


9S-4idJ 1 IW 




901 


47 1 1 a 




903 




40 




903 


6688 


5 1 


48 


904 


6168 








Gil 01 


48 


905 


6649 


6697 


6745 


6793 


6840 


48 


901 




7176 


73 4 


7273 


73 (, 9 


48 


907 


76U7 


7855 


7703 


7751 


7704' 038 


48 


SOS 


808r 


en4 


8181 


82 9 


827/ 9 


48 


909 


8564 


8612 


8659 


8707 


87 m 


48 


Bio 










921 (171 




911 


9518 


9566 


9614 


96ri 






91S 






qo 


139 


18 1 4 3 




913 


960471 


19 


0566 


01 H 1 1 44 




914 


094 


Oq94 


1041 


1 114 




915 


14 


1469 


1611, 1 1 Ifl48 


47 


Sl<t 


18J5 


194 


19JI ! 


47 


917 


36 


117 


346ll 1 J'i 


47 


918 


S4 


S)l) J( II J 69 


47 


919 


331 


^ ' _ iiiili 


47 


sao 


96FSS 


UIH 41b^U2U 


"47 


9E1 


4 til 1 0U14637 lb84 


47 


9S2 


47d! 


1 Obi 6108 5150 


47 


923 


6 


11 ^5625 


47 


924 


667 


1 1 1 i^lnias 


47 


835 


614 


61 t i 1 1 IW4 


47 


9S6 


601 ; 


6658 6 05 F75 C m 


47 


927 




71 7 7Ui 7 7 01 


47 


938 




7M1 764 7( 7 1 Jbl 


47 


BSD 

935 






47 

47 


96S1 1 








93! 




9 29 


9276 








932 


911 1 li 


9695 




9789 


9831 




933 


98 11' 


16: 


207 


364 


300 




934 






0672 


0719 


0765 


46 


93S 


0812 lO (II|UJ5i|l)l 104' 


1091 


1137 


1183 


1339 


46 


936 


1 76 13 2lu J 141 14 I 1 Ue 


15'v 


160 


1647 


1693 




937 


1740 17861 183 1879 1925 197 




206. 


2110 


S15? 


46 


939 


2201 2 49 q-i 2342 188 243 






173 


26 IS 


46 




666 71 1 S' 80*1 2B 12897 943lS380 3031 308S 46 


IT 


1 1 1 2 1 3 


'l6~ 


1 5 1 1 7 1 8 1 9 1 D. 














■-D1 











16 


▲ TABLE OF L0OARITH!ffS FKOX f 


TO lOfiOO 


► 




N. 


l|2|3|4 dr|6|7 8i9 


D. 


940 


973128 


3174 


3220 


3266 


3313 


3359 34051 


3451 3497 


3543 


46 


941 


3590 


3636 


36S2 


3728 


3774 


3820 


3866 


3913 


3959 


4005 


46 


942 


4051 


4097 


4143 


4189 


4236 


4281 


4327 


4374 


4420 


4466 


46 


943 


4512 


4558 


4604 


4650 


4696 


4742 


4788 


4^34. 


4880 


4926 


46 


944 


4972 


5018 


5064 


5110 


5156 


5202 


5248 


6294 


5340 


5386 


46 


945 


5432 


5478 


5524 


5570 


5616 


5662 


5707 


6753 


6799 


5845 


46 


946 


5891 


5937 


5983 


6029 


6076 


6121 


6167 


6212 


6258 


6304 


46 


947 


6350 


6396 


6442 


6488 


6533 


6579 


6625 


6671 


6717 


6763 


46 


948 


6808 


6854 


6900 


6946 


6992 


7037 


7083 


7129 


7175 


7220 


46 


949 
950 


7266 


7312 
7769 


7358 
7815 


7403 
7861 


7449 
7906 


7496 
7952 


7541 
7998 


7586 
8043 


7632 
8089 


7678 
8135 


46 
46 


977724 


951 


8181 


8226 


8272 


8317 


8363 


8409 


8454 


8500 


8546 


8591 


46 


952 


8637 


8683 


8728 


8774 


8819 


8866 


8911 


8956 


9002 


9047 


46 


953 


9093 


9138 


9184 


9230 


9275 


9321 


9366 


9412 


9457 


9503 


46 


954 


954S 


9594 


9639 


9685 


9730 


0776 


9821 


9867 


9912 


9958 


46 


955 


980003 


0049 


0094 


0140 


0185 


0231 


0276 


0.322 


0367 


0412 


45 


956 


0458 


0503 


0549 


0594 


0640 


0685 


0730 


0776 


0821 


0867 


45 


957 


0912 


0957 


1003 


1048 


1093 


1139 


1184 


1229 


1275 


1320 


45 


958 


1366 


1411 


1456 


1501 


1547 


1592 


1637 


1683 


1728 


1773 


45 


959 
960 


1819 


1864 
2316 


1909 
2362 


1954 
2407, 


2000 
2452 


2045 
2197 


2090 
2543 


2135 

2588 


2181 
2633 


2226 
2678 


45 
45 


982271 


961 


2723 


2769 


2814 


2859 


2904 


2949 


2994 


3040 


3085 


3130 


45 


962 


3175 


3220 


3265 


3310 


3356 


3401 


3446 


3491 


3536 


3581 


45 


963 


3626 


3671 


3716 


3762 


3807 


3852 


3897 


3942 


3987 


4032 


45 


964 


4077 


4122 


4167 


4212 


4257 


4302 


4347 


4392 


4437 


4482 


45 


965 


4527 


4572 


4617 


4062 


4707 


4752 


4797 


4^42 


4887 


4932 


45 


966 


4977 


5022 


5067 


5112 


5157 


5202 


5247 


5292 


5337 


5382 


45 


967 


5420 


5471 


5516 


5561 


5606 


5651 


5696 


5741 


5786 


5830 


45 


968 


5875 


5920 


5965 


6010 


6055 


6100 


6144 


6189 


6234 


6279 


45 


969 
970 


6324 


6369 
6817 


6413 
6861 


6458 
6906 


6503 
6951 


6548 
6996 


6593 
7040 


6637 
7085 


6682 
7130 


6727 
7175 


45 
45 


986772 


971 


7219 


7264 


7309 


7353 


7398 


7443 


7488 


7532 


7577 


7622 


45 


972 


7666 


7711 


7756 


7800 


7845 


7890 


7934 


7979 


8024 


8068 


45 


973 


8113 


8157 


8202 


8247 


8291 


8336 


8381 


8425 


8470 


8514 


45 


974 


8559 


8604 


8848 


8693 


8737 


8782 


8826 


8871 


8916 


89C0 


45 


975 


9005 


9049 


9094 


9138 


9183 


9227 


9272 


9316 


9361 


9405 


45 


976 


9450 


9494 


9539 


9583 


9628 


9672 


9717 


9701 


9806 


9850 


44 


977 


9895 


9939 


9983 


..28 


..72 


.117 


.161 


.206 


.2.50 


.294 


44 


978 


990339 


0383 


0428 


0472 


0516 


0561 


0605 


0650 


0694 


0738 


44 


979 
9S0 


0783 


0827 
127t) 


0871 
1315 


0916 
1359 


0960 
1403 


1004 

1448 


1049 
1492 


1093 
1.536 


1137 

1580 


1182 
1625 


44 
44 


991226 


981 


1669 


1713 


1758 


1802 


1846 


1890 


1935 


1979 


2023 


20C7 


44 


982 


2111 


2156 


2200 


2244 


2288 


2333 


2377 


2421 


2465 


2509 


44 


983 


2554 


2598 


2642 


2686 


2730 


2774 


2819 


2S63 


2907 


2951 


44 


984 


2995 


3039 


3083 


3127 


3172 


32161 3260 


3304 


3348 


.3392 


44 


985 


3436 


3480 


3524 


3568 


3613 


3657 


3701 


3745 


3789 


3833 


44 


986 


3877 


3921 


3965 


4009 


4053 


4097 


4141 


4185 


4229 


4273 


44 


987 


4317 


4361 


4405 


4449 


4493 


4537 


4581 


4625 


4669 


4713 


44 


988 


4757 


4801 


4845 


4889 


4933 


4977 


5021 


.5065 


5108 


5152 


44 


989 
990 


6196 


5240 
5679 


5284 
5723 


5328 
5767 


5372 
5811 


5^16 

5854 


5460 
.5898 


5504 
5942 


6547 

5988 


5591 
6030 


44 
44 


995635 


!99l 


6074 


6117 


6161 


6205 


6249 


6293 


6337 


6380 


6424 


6468 


44 


992 


6512 


6555 


6599 


6643 


6687 


6731 


6774 


6818 


6862 


6906 


44 


!993 


6949 


6993 


7037 


7080 


7124 


7168 


7212 


7255 


7299 


7343 


44 


(1^94 


7386 


7430 


7474 


7517 


7561 


7605 


7648 


7092 


7736 


7779 


44 


• 995 


7823 


7867 


7910 


7951 


7998 


8041 


8085 


8129 


8172 


8216 


44 


i996 


8259 


8303 


8347 


8390 


8434 


8477 


8521 


8564 


8608 


8652 


44 


997 


8695 


8739 


8782 


8826 


8869 


8913 


89.56 


9000 


9043 


9087 


44 


998 


9131 


9174 


9218 


92-01 


9305 


9348 


9392 


9435 


9479 


9522 


44 


999 


9565 


9609 


9652 9rj:)6 


9739 


9783 


9820 


9870 


9913 


9957 


43 


N. 


|l|2|3|4|5l6|7|8|P« 


D. 



A TABLE 



OF 



LOGARITHMIC 



8INES AND TANGENTS, 



FOR BYERT 



DEGREE AND MINUTE 



OF THE QUADRANT. 



N. B The minutes in the lefl-hand column of each pa^, 
mcreasing downwards, belong to the degrees at the top ; and 
ihose increasing upwards, in the right>hand column, belong to 
the degrees below. 



18 


(0 Def^rae.) a table of uoa^amaao 




~ 


Sliie 


D. 


i:,«i,.« 1 n. 


T„* 


1,. 


c... 1 1 


= 


O.OOIIOIIO 




lOMmin 




0. OOOOOO 






"Bo 




6.4r.3736 


601 rn 




00 


6.4G3736 


501717 


13.536374 






764756 


293485 


ooom 


00 






235344 


58 




940847 


80833! 


OOOflOO 


00 






059153 






7.065786 


1B1617 


oooooo 


00 


7.065786 


161517 


13.934214 






163606 


131968 


OOOOOl 


OO 


163696 


I319B[ 


837304 






!Mi877 


111676 










758133 






808834 


96663 






308825 




631175 






366816 


85364 






306817 




633183 






417988 


70363 






417970 




682039 




10 

r 


463726 


68988 




oT 


463737 


68988 


536373 


50 

49 


7.6i;51!8 


62931 




7.605130 


6398! 


13.494880 


la 


642906 


67938 


99999; 


01 


643909 


57933 


4570 J 1 


8 


1 


677668 


53641 




01 


677673 


364| 


423338 




14 


G09853 


49938 


99990 


0! 


609867 




390143 




16 




46714 




01 


639830 


6716 


360180 


' fi 


1 


B67845 


43881 


999996 


01 


667849 


3882 


3.13151 


44 


17 


694173 


41373 


999995 


01 


604l7il 


1373 


305S31 


43 


1 




39135 




01 


71900E 


9136 


380997 


3 


1 


742477 


37137 


99999J 


01 


743484 


37128 


S57;i6 


1 


SO 

s 


7B4754 


36316 


9.999993 


01 
01 


764761 


35136 


335239 



"9 


7.78694! 
SOB 146 


"33Sw 


7.786951 


"raSTs 


13.S14II49 




33175 


99999! 


01 


806155 


33178 




38 




S36431 


30805 


999990 


01 


8254B0 


30S06 


174540 


7 


S4 


843934 


39547 


999989 


02 


843944 


29649 


156056 


30 




861 663 


38388 


99998S 


02 


861674 


38390 


138336 


5 


SB 


878666 


87317 


099988 


03 


878708 


37318 


121293 






895086 


36333 


099D87 


02 


895099 


36336 


104901 


3 


38 


91087 


3639^ 


999980 


02 


010894 


3.'i401 


089106 




S9 


9EG119 


34538 


999985 


02 


926134 


34 MO 


073868 




SO 


940843 




09D983 


02 
03 


04U858 


33735 


0.59143 


30 


7.9B50SS 


32980 




7.9.S9100 




IS.O4490I) 




903870 


22373 


9999S1 


03 


968889 




03111 




3a 


983333 


SI60S 


099980 


03 


93335.1 


21610 


01774 


27 


84 


9BS19B 


30981 


899979 


03 


995319 




00478 


S6 


B6 


8.0077B7 


20390 




03 


e. 007809 




11.99319 


36 




oaoosi 


9831 




03 


030045 




97995. 




3T 




9302 


099975 


03 


031945 


1B305 


96805 


33 


88 


048501 


8801 




02 


043527 


19803 


95647 


23 


80 




8335 




02 


054809 


18337 


945191 


21 


M 


06677H 




999^71 


02 


065806 


17874 


934194 


30 


41 


8.070600 


"TtmT 


9.999368 




8.070531 


17444 


11.933469 


!9 


4a 


086985 


17031 






086997 


17034 


913003 


IS 


43 


0971 S3 


16039 




03 


0U7317 


1964! 


903783 


17 


44 




36265 


999964 


03 




16268 




16 


4S 


I!69Sa 


15908 


990963 


03 


U696.S 


1591 


BS303'; 


16 


4fi 


136471 


15566 


999961 


09 


136510 


15568 


873490 




47 


136810 


16338 




03 


135961 


15341 


864149 


13 


48 


1449B3 


14934 


999958 




144996 


14it27 


855004 




4S 


153907 


14633 


999958 




IS395S 


14B27 


846048 




fiO 
61 


163681 


14333 


9999S4 


03 
03 


163727 


14.13 




^ 


8.171SS0 


14054 


9.999953 


8.171328 


1405 


11 83867! 


6S 


179713 


13786 


999950 


03 


179763 


13790 






63 


187985 


13639 


999948 


03 


188036 


13532 


8U064 




64 


196103 


13280 


999946 


03 


196156 


13384 


803844 




65 


304070 


13041 


999944 


;3 


204136 


13044 


795874 


6 


ee 


311896 


12810 


999943 


'4 


311963 


13814 


788047 




67 


319581 


12587 


999940 


04 


219641 


13590 


78U359 




ee 


337134 


13372 


999938 


04 


227196 


1337b 


773805 




69 


334557 


12164 




04 


234621 


13168 


765379 




60 


341855 


llyfiS 


99993^ 


04 


241921 


11987 


76S079 





" 


C„.. 




Sin. 1 


(JoUiJif. 




-- !«■ 



mm AND TANGSICT8. (1 Degree.) 



19 



M. 


Siiie 


D. 


Cosine 1 D. 


1 Tang. 


D 


Cotang. 


T 


8.241855 


11963 


9.999934 


04l 


8.241921 


11967 


11.758079 


60 


1 


249033 


11768 


999932 


04 


249102 


11772 


750898 


f-^ 


2 


256094 


11580 


999929 


04 


256166 


11584 


743835 


d8 


8 


263042 


11398 


999927 


04 


263115 


11402 


736885 


57 


4 


269881 


11221 


999925 


04 


269956 


11225 


730044 


56 


5 


276614 


11050 


999922 


04 


276691 


11054 


723309 


55 


6 


283243 


10883 


999920 


04 


283323 


10887 


716677 


54 


7 


289773 


10721 


9^9918 


04 


289856 


10726 


710144 


53 


8 


296207 


10565 


999915 


04 


296292 


10570 


703708 


52 


9 


302546 


10413 


999913 


04 


302634 


10418 


697366 


51 


10 

11 


308794 
8.314954 


10266 


999910 


04 
04 


308884 


10270 


691116 


50 


10122 


9.999907 


8.315046 


10126 


11.684954 


49 


12 


321027 


9982 


999906 


04 


321122 


9987 


678878 


48 


13 


327016 


9847 


999902 


04 


327114 


9851 


672886 


47 


14 


332924 


9714 


999899 


05 


333026 


9719 


666975 


46 


15 


338753 


9586 


999897 


05 


338S56 


9590 


661144 


45 


16 


344504 


9460 


999894 


05 


344610 


9465 


655390 


44 


17 


350181 


9338 


999891 


05 


350289 


9343 


649711 


43 


18 


355783 


9219 


999888 


05 


355895 


9224 


644105 


42 


19 


361315 


9103 


999885 


05 


361480 


9108 


638570 


41 


20 
21 


366777 


8990 


999882 


05 
05 


366895 


8995 


633105 


40 


8.372171 


8880 


9.999879 


8.372292 


8886 


11.627708 


39 


22 


077499 


8772 


999876 


05 


377622 


8777 


622376 


38 


23 


382762 


8667 


999873 


06 


382889 


8672 


617111 


37 


24 


387962 


8564 


999870 


05 


388092 


8570 


611908 


36 


25 


303101 


8464 


999867 


05 


393234 


8470 


606766 


35 


26 


898179 


6366 


999864 


05 


398316 


8371 


601686 


34 


27 


403199 


8271 


999861 


06 


403338 


8276 


596662 


33 


28 


408161 


8177 


999858 


05 


408304 


8182 


591696 


32 


29 


413068 


8086 


999854 


05 


413213 


8091 


586787 


31 


ao 

31 


417919 


7996 


999851 


06 
06 


418066 


8002 


581932 


30 


8.422717 


7909 


9.999848 


8.422869 


7914 


11.577131 


29 


32 


427462 


7823 


999844 


06 


427618 


7830 


572382 


28 


33 


432166 


7740 


999841 


06 


432316 


7745 


567686 


27 


34 


436800 


7657 


999638 


06 


436962 


7663 


563036 


26 


35 


441394 


7677 


999834 


06 


441560 


7583 


568440 


25 


36 


445941 


7499 


999831 


.06 


446110 


7505 


553890 


24 


37 


450440 


7422 


999827 


06 


460613 


7428 


549387 


23 


38 


454893 


7346 


999823 


06 


465070 


7352 


544930 


22 


39 


459301 


7273 


999820 


06 


469481 


7279 


540519 


21 


40 
41 


463065 


7200 


999816 
9.999812 


06 
06 


463849 


7206 


536161 


20 


8.467985 


7129^ 


8.468172 


7135 


11.531826 


19 


42 


4722G3 


7060 


999609 


06 


472454 


7066 


527546 


18 


43 


476498 


6991 


999805 


06 


476693 


6998 


523307 


17 


44 


480693 


6924 


999801 


06 


480892 


6931 


519106 


16 


45 


484848 


6859 


999797 


07 


485050 


6865 


514950 


15 


46 


489963 


6794 


999793 


07 


489170 


6801 


510830 


14 


47 


493040 


6731 


999790 


07 


493250 


6738 


506750 


13 


48 


497078 


6669 


999786 


07 


497293 


6676 


502707 


12 


49 


5010S0 


6608 


999782 


07 


501298 


6616 


498702 


11 


50 
51 


505045 


6548 


999778 


07 
07 


505267 


6555 


494733 


10 


8.508974 


6489 


9.999774 


8.509200 


6496 


11.490800 


9 


52 


512867 


6431 


999769 


07 


513098 


6439 


486902 


8 


53 


516726 


6375 


999765 


07 


516961 


6382 


483039 


7 


54 


520551 


6319 


999761 


07 


520790 


6326 


479210 


6 


65 


524343 


6264 


999757 


07 


524586 


6272 


475414 


5 


56 


528102 


6211 


999753 


07 


528349 


6218 


471651 


4 


57 


531828 


6158 


999748 


07 


532080 


6165 


467920 


8 


58 


535523 


6106 


999744 


07 


535779 


6113 


464221 


% 


59 


539186 


6055 


999740 


07 


539447 


6062 


46AUM 




60 


542819 


6004 


9997.36107 


543084 


6012 


4- 




Li 


Cosine 1 




1 Bine 1 


Ootang. 




pB 



abDegnaa 



80 


(2 Degrees.) a 


TABLE OF LOOASITHHIO 




nr 


0|pe 1 


D. 1 


Cosine | D. 


1 Tanf. 


1 D. 


Cotanf;. | | 





8.542819 


6004 


9.999736 


07 


8.643084 


6012 


11.456916 


60 


1 


546422 


5955 


999731 


07 


546691 


5962 


453309 


69 


2 


549995 


5906 


999726 


07 


650268 


5914 


449732 


68 


3 


653539 


5858 


999722 


08 


653817 


6866 


446183 


67 


4 


657054 


5811 


999717 


08 


557336 


6819 


442664 


66 


5 


560540 


5765 


999713 


08 


660828 


6773 


439172 


66 


6 


563999 


6719 


999708 


08 


664291 


5727 


435709 


64 


7 


667431 


6674 


999704 


08 


567727 


6682 


432273 


63 


8 


670836 


5630 


999699 


08 


571137 


56.38 


428863 


62 


9 


674214 


5687 


999694 


08 


574520 


5595 


425480 


61 


10 

11 


677666 


6544 


999689 


08 
08 


577877 


5552 


422123 


60 
49 


8.580892 


6502 


9.999685 


8.581208 


6510 


11.418792 


12 


684193 


6460 


999680 


08 


584514 


6468 


415486 


48 


13 


587469 


6419 


999675 


08 


587795 


6427 


412205 


47 


14 


690721 


5379 


999670 


08 


591051 


6387 


408949 


46 


16 


593948 


5339 


999665 


08 


594283 


5347 


405717 


46 


16 


597152 


6300 


999660 


08 


697492 


6308 


402508 


44 


17 


600332 


6261 


999655 


08 


600677 


6270 


399323 


43 


18 


603489 


6223 


999650 


08 


603839 


5232 


396161 


42 


19 


606623 


6186 


999645 


09 


606978 


5194 


393022 


41 


20 
21 


609734 


5149 


999640 


09 
09 


610094 


6158 


389906 


40 
39 


8.612823 


5112 


9.999635 


8.613189 


5121 


11.386811 


22 


615891 


5076 


999629 


09 


616262 


5085 


383738 


38 


23 


618937 


5041 


999624 


09 


619313 


5050 


380687 


37 


24 


621962 


6006 


999619 


09 


622343 


5015 


377657 


36 


25 


624965 


4972 


999614 


09 


625352 


4981 


374648 


35 


26 


627948 


4938 


999608 


09 


628340 


4947 


371660 


34 


27 


630911 


4904 


999603 


09 


631308 


4913 


368692 


33 


28 


633854 


4871 


999597 


09 


634256 


4880 


365744 


32 


29 


636776 


4839 


999592 


09 


637184 


4848 


362816 


31 


30 
31 


639680 


4806 


999586 


09 
09 


640093 


4816 


359907 


30 
29 


8.642563 


4776 


9.999581 


8.642982 


4784 


11.357018 


32 


645428 


4743 


999575 


09 


645853 


4753 


364147 


28 


33 


648274 


4712 


999570 


09 


648704 


4722 


351296 


27 


34 


651102 


4682 


999564 


09 


651537 


4691 


348463 


26 


35 


653911 


4652 


999558 


10 


654352 


4661 


346648 


25 


36 


656702 


4622 


999553 


10 


. 657149 


4631 


342851 


24 


37 


659475 


4592 


999547 


10 


659928 


4602 


340072 


23 


38 


6G2230 


4563 


999541 


10 


662689 


4573 


337311 


22 


39 


664968 


4535 


999535 


10 


665433 


4544 


334567 


21 


40 
41 


667689 


4506 
4479 


999529 


10 
10 


668160 


4526 


331840 


20 
19 


8.670393 


9.999524 


8.670870 


4488 


11.329130 


42 


673080 


4451 


999518 


10 


673563 


4461 


326437 


18 


43 


676761 


4424 


999512 


10 


676239 


4434 


323761 


17 


44 


678405 


4397 


999506 


10 


678900 


4417 


321100 


16 


45 


681043 


4370 


999500 


10 


681544 


4380 


318456 


16 


46 


683665 


4344 


999493 


10 


684172 


4354 


315828 


14 


47 


686272 


4318 


999487 


10 


686784 


4328 


313216 


13 


48 


688863 


4292 


999481 


10 


689381 


4303 


310619 


12 


49 


691438 


4267 


999475 


10 


691963 


4277 


308037 


11 


50 
51 


693998 


4242 


999469 


10 
11 


694529 
8.697081 


4252 


305471 


10 
9 


8.696343 


4217 


9.999463 


4228 


11.302919 


52 


699073 


4192 


999456 


11 


699617 


4203 


300383 


8 


53 


701589 


4168 


999450 


11 


702139 


4179 


297861 


7 


54 


704090 


4144 


999443 


11 


704646 


4155 


295364 


6 


55 


706577 


4121 


999437 


11 


707140 


4132 


292860 


5 


66 


709049 


4097 


999431 


11 


709618 


4108 


290382 


4 


67 


711507 


4074 


999424 


11 


712083 


4085 


287917 


3 


58 


713952 


4051 


999418 


U 


714534 


4062 


285465 


2 


59 


716383 


4029 


999411 


11 


716972 


4040 


283028 


1 


60 


718800 


4006 


999404 


11 


719396 


4017 


280504 




Coeine 


1 


Sine 1 


Cotanc. 


1 


Tang. IM. 



«7 Degreea. 





« 


HEB ATD TAHHEifn. {3 Vegnm. 




fl 




M. 


gi»e 1 


D. 


(•-rt\aii 1 D. 


T„. 


D. 


r.«.. 




tf 


S.7IH(i00 


4006 


9;809W-r 


11 


8.71939 


4017 


11.280604 


-60 






721204 


3934 


999398 


11 


731806 


3995 


S78104 


59 




i 




3063 


099391 


11 


734304 


3974 


376796 


68 




s 


735972 




999.384 


11 


736688 


3953 


2734 U 


67 




4 


738337 


391 B 


999378 


11 


738959 


3930 


371041 


56 




6 


7306B8 


3S9B 


999371 


U 


731317 


8909 


3686B3 


55 




6 


733087 


3877 


99036-1 


IE 


733663 


3889 


S66337 


54 




7 


735364 


3357 


99B357 


13 






354004 


63 




8 


7376b7 


3836 


999350 


13 




3348 


361683 






9 


739969 


3816 


999343 


13 


740636 


3837 


369374 






ID 

n 


743389 


3796 
3776 


999330 


13 

12 


742933 


3807 


2S7078 


50 

49 




8.744536 


3.999329 


8.745207 


3787 


uTasSm 




1) 


746803 


3756 


999322 


13 


747479 


3768 


253631 


48 




13 


74906 


3737 


999315 




749740 


3749 


350360 


47 




14 


7G1S9 


3717 


999308 


13 


751989 


3729 


348011 


46 




15 


763528 


3698 


999301 


13 


754337 


3710 


346773 


46 




16 


755747 


3679 


999394 




756453 


3693 


343547 


44 




17 


757956 


3661 


999386 


13 


758668 


3673 


341333 


43 




IB 


760151 


3643 


999379 


12 


760872 


3655 


339138 


42 




19 


763337 


3634 


999373 




763065 


3636 


336935 


41 




31 


764511 


3606 


999365 


12 


765346 


3618 
3600 


234754 
11,383583 


40 
39 




8.766676 




9.990367 


8.767417 




sa 


768338 


3570 


9992.'iO 


; 


769578 


3583 


330433 


38 




33 


770970 


3553 


999343 




771737 


3565 


338273 






S4 


773101 


3535 


999335 




7T3S66 


3548 


326134 


36 




as 


775333 


3518 


999337 




77B995 


3531 


324005 


35 




an 


777S33 


3501 


999330 




778114 


3514 


S2189(, 


34 




27 


770434 


3484 


999213 




780222 


3497 


219778 






as 


781534 


3467 


9BB3D5 


t 


782330 


3480 


317B80 






39 


783605 


3451 


909197 


3 


784408 


3464 


315693 






so 

3l 


785675 


343! 


999189 


i; 


786486 


3447 


313514 






8.787736 


3418 


9.99918: 




3431 


U. 2114-16 




31 


789787 


3403 


999174 




"79061! 


3414 


509387 






33 


TBisas 


33S6 


999166 




793662 




X07338 






S4 


703^59 


3370 


999158 


i: 


794701 




305399 






3& 




3354 


999160 


i: 


796731 


3368 


303369 






36 


797884 




900142 




799753 


3353 


201348 






37 


7999B7 




9B9134 




800763 


3337 


199337 






39 




U308 


99913B 


i; 






197335 






39 


S03S76 




999118 




804768 




I 95343 






iO 
41 


805353 




9991 Ifl 


13 


806743 


3293 


193358 


; 




8.807819 


3363 


9.99910! 


8.808717 


3378 






43 


809777 


3349 


999094 


14 


810683 


3363 


89317 


; 




43 


811736 


3334 


99908 


14 


813041 


3348 


e-/369 






44 


813667 


3219 


99907 


14 


814589 


3333 


85411 


. 




46 


815599 


3 OS 


99906 


14 


816539 


3319 


83471 


15 




46 


817522 


3 Bl 


99906 


14 


818461 


3305 


81539 






47 




3 77 


99905 


14 


820384 


3191 


796 ir 


13 




48 


821341 


3 63 


999044 


14 


833398 


3177 


77703 


13 




49 


823340 


3 49 


99903 


14 


824305 


3163 








SO 


835130 


3 35 


99903 


14 
14 


836103 


3150 




to 

"9 




8.837011 




9.99901 


8.827992 


"alir 


TT:~730iw 




sa 


828884 


3108 


99901 


14 


839874 


3133 


701 2(- 


8 




63 


830749 


3095 


09900 


14 


83174S 


3110 


08253 


7 




64 


833607 


3082 




14 


833613 


3096 


66387 


6 




65 


834461 


3069 


99898 


1. 


83547 


3083 


64520 


6 




66 




3056 


99897 






3070 


62679 


4 




67 


838131 


3043 




li 


83916: 


3057 


60837 






68 




3030 


99395 


15 


840998 


3045 


169002 


3 




69 


84177' 






h 


843835 


3033 


167175 






60 


843580 


3000 


09894 




844644 


3019 


155356 







: ..;.» , 


Sl..e 1 


-SS-^IT- 




T.,„. |«. 





22 


(4 DegreesO a 


TABLE OF LOOABITHMIO 




M. 


Sine 


1 D. 1 


Cosine | D. 


Tang. 


D. 


Coung. 1 1 





8.843685 


3006 


9.998941 


15 


8.844644 


3019 


11.165356 


60 


1 


845387 


2992 


998932 


15 


846455 


3007 


163545 


69 


2 


847183 


2980 


998923 


15 


848260 


2996 


161740 


68 


3 


848971 


2967 


998914 


15 


850057 


2982 


149943 


57 


4 


850761 


2966 


998905 


16 


851846 


2970 


148164 


66 


5 


862526 


2943 


998896 


15 


853628 


2958 


146372 


55 





864291 


2931 


998887 


16 


856403 


2946 


144697 


64 


7 


866049 


2919 


998878 


16 


857171 


2936 


142829 


53 


8 


857801 


2907 


998869 


16 


868932 


2923 


141068 


52 


9 


859646 


2896 


998860 


15 


860686 


2911 


139314 
137567 


61 


10 

11 


861283 


2884 


998851 


16 
16 


862433 


2900 


60 
49 


8.863014 


2873 


9.998841 


8.864173 


2888 


11.135827 


12 


864738 


2861 


998832 


16 


865906 


2877 


134094 


48 


13 


866455 


2850 


998823 


16 


867632 


2866 


132368 


47 


14 


868166 


2839 


998813 


16 


869351 


2864 


130649 


46 


16 


869868 


2828 


998804 


16 


871064 


2843 


128936 


46 


16 


871665 


2817 


998795 


16 


872770 


2832 


127230 


44 


17 


873265 


2806 


998785 


16 


874469 


2821 


125531 


43 


18 


874938 


2796 


998776 


16 


876162 


2811 


123838 


42 


19 


876615 


2786 


998766 


16 


877849 


2800 


122161 


41 


20 
21 


878285 


2773 


998757 


16 
16 


879529 


2789 


120471 


40 
39 


8.879949 


2763 


9.998747 


8.881202 


2779 


11.118798 


22 


881607 


2762 


998738 


16 


882869 


2768 


117131 


38 


23 


883258 


2742 


998728 


16 


884530 


2768 


115470 


37 


24 


884903 


2731 


998718 


16 


886185 


2747 


113815 


86 


26 


886642 


2721 


998708 


16 


887833 


2737 


112167 


35 


26 


888174 


2711 


998699 


16 


889476 


2727 


110624 


34 


27 


889801 


2700 


998689 


16 


891112 


2717 


108888 


33 


28 


891421 


2690 


998679 


16 


892742 


2707 


107258 


32 


29 


893036 


2680 


998669 


17 


894366 


2697 


105634 


31 


30 

31 


894643 


2670 


998669 


17 
17 


895984 


2687 


104016 


30 
29 


8.890246 


2660 


9.998649 


8.897590 


2677 


11.102404 


32 


897842 


2661 


998639 


17 


899203 


2667 


100797 


28 


33 


899432 


2641 


998629 


17 


900803 


2658 


099197 


27 


34 


901017 


2631 


998619 


17 


902398 


2648 


097602 


26 


35 


902596 


2622 


998609 


17 


903987 


2638 


096013 


26 


38 


904169 


2612 


998599 


17 


905570 


2629 


094430 


24 


37 


905736 


2603 


998589 


17 


907147 


2620 


092853 


23 


38 


907297 


2593 


998578 


17 


908719 


2610 


091281 


22 


39 


908853 


2584 


998568 


17 


910285 


2601 


089715 


21 


40 
41 


910404 


2575 


998558 


17 
17 


911846 


2592 


088154 


20 
19 


8.911949 


2566 


9.998548 


8.913401 


2583 


11.086599 


42 


913488 


2556 


998537 


17 


914951 


2574 


085049 


18 


43 


9150J.2 


2647 


998627 


17 


916495 


2566 


083505 


17 


44 


916550 


2538 


998516 


18 


918034 


2556 


081966 


16 


45 


918073 


2529 


998506 


18 


919568 


2647 


080432 


16 


46 


919591 


2520 


998495 


18 


921096 


2538 


078904 


14 


47 


921103 


2512 


998485 


18 


922619 


2530 


077381 


13 


48 


922610 


2503 


998474 


18 


924136 


2521 


075864 


12 


49 


924112 


2494 


998464 


18 


925649 


2512 


074351 


11 


50 
51 


925609 


2486 


998453 
9.998442 


18 
18 


927156 


2503 


072844 


10 
9 


8.927100 


2477 


8.928658 


2495 


11.071342 


52 


928587 


2469 


998431 


18 


930155 


2486 


069845 


8 


53 


930068 


2-160 


998421 


18 


931647 


2478 


068353 


7 


54 


931644 


2452 


998410 


18 


933134 


2470 


066866 


6 


56 


933015 


2443 


998399 


18 


934616 


2461 


065384 


5 


56 


934481 


2435 


998388 


18 


936093 


2453 


063907 


4 


57 


935942 


2427 


998377 


18 


937565 


2445 


062435 


3 


68 


937398 


2419 


998366 


18 


939032 


2437 


060968 


2 


59 


938850 


2411 


998356 


18 


940494 


2430 


059506 


1 


60 


940296 


2403 


998344 


18 


941952 


2421 


058048 








Cosine 




Sine 


1 Cotang. 




1 Tang. M. 1 



85 Degrees. 





ninu AND TANoxivTs. (5 Degrrees.} 


23 


pn 


Bine 


D. 1 


Coeine | D. 


1 Tang. 


1 a 


1 Comng. 1 1 





8.940296 


24U3 


9.9983441 


19 


S. 941952 


2421 


11.0580481 60 1 


1 


941738 


2394 


998333 


19 


943404 


2413 


056596 


69 


s 


943174 


2387 


998322 


19 


944852 


2405 


056148 


58 


3 


944606 


2379 


998311 


19 


946295 


2397 


053705 


57 


4 


946034 


2371 


998300 


19 


947734 


2390 


052286 


56 


6 


947466 


2363 


998289 


19 


949168 


2382 


050832 


65 





948874 


2355 


998277 


19 


950597 


2374 


049403 


54 


7 


950287 


2348 


998266 


19 


952021 


2366 


047979 


63 


8 


951696 


2340 


998255 


19 


953441 


2360 


046559 


62 


9 


953100 


2332 


998243 


19 


954856 


2351 


046144 


61 


10 
11 


954499 


2325 


998232 


19 
19 


956267 


2344 


043733 


50 
49 


8.955894 


2317 


9.998220 


8.957674 


2337 


11.042326 


12 


957284 


2310 


998209 


19 


959075 


2329 


040925 


48 


13 


958670 


2302 


998197 


19 


960473 


2323 


039527 


47 


14 


960052 


2295 


998186 


19 


961866 


2314 


038134 


46 


lf> 


961429 


2288 


998174 


19 


963255 


2307 


036745 


46 


16 


962801 


2280 


998163 


19 


964639 


2300 


036361 


44 


17 


964170 


2273 


998151 


19 


966019 


2293 


033981 


43 


18 


965534 


2266 


998139 


20 


967394 


2286 


032606 


42 


19 


966893 


2259 


998128 


20 


988766 


2279 


031234 


41 


20 
21 


968249 


2252 


998116 
9.998104 


20 
20 


970133 


2271 


029867 


40 
39 


8.969600 


2244 


8.971496 


2265 


11.028504 


22 


970947 


2238 


998092 


20 


972855 


2257 


027145 


38 


23 


972289 


2231 


998080 


20 


974209 


2251 


025791 


37 


24 


973628 


2224 


998088 


20 


975560 


2244 


024440 


36 


25 


974962 


2217 


998058 


20 


976906 


2237 


023094 


36 


26 


976293 


2210 


993044 


20 


978248 


2230 


021762 


34 


27 


977619 


2203 


998032 


20 


979586 


2223 


020414 


33 


28 


978941 


2197 


99S020 


20 


980921 


2217 


019079 


32 


29 


980259 


2190 


998008 


20 


982251 


2210 


017749 


31 


30 
31 


981573 


2183 


997998 
9.997984 


20 
20 


983577 


2204 


016423 


30 

29 


8.982883 


2177 


8.984899 


2197 


11.015101 


32 


984189 


2170 


997972 


20 


986217 


2191 


013783 


28 


33 


985491 


2163 


997959 


20 


987632 


2184 


012468 


27 


34 


986789 


2157 


997947 


20 


988842 


2178 


011158 


26 


35 


988083 


2150 


997935 


21 


990149 


2171 


009851 


26 


36 


989374 


2144 


997922 


21 


991451 


2165 


008549 


24 


37 


990660 


2138 


997910 


21 


992750 


2158 


007250 


23 


38 


991943 


2131 


997897 


21 


994045 


2152 


005955 


22 


39 


993222 


2125 


997885 


21 


995337 


2146 


004663 


21 


40 
41 


994497 
8.995768 


2119 


997872 
9.997880 


21 
21 


996624 


2140 


003376 


20 
19 


2112 


8.997908 


2134 


11.002092 


42 


997036 


2106 


997847 


21 


999188 


2127 


000812 


18 


43 


998299 


2100 


997835 


21 


9.000465 


2121 


10.999.535 


17 


44 


999560 


2094 


997322 


21 


001738 


2115 


998262 


16 


45 


9.000816 


2087 


997809 


21 


003007 


2109 


996993 


15 


46 


002069 


2032 


997797 


21 


004272 


2103 


995728 


14 


47 


003318 


2078 


997784 


21 


005534 


2097 


994466 


13 


48 


004563 


2070 


997771 


21 


006792 


2091 


993208 


12 


49 


005805 


2084 


997758 


21 


008047 


2085 


991953 


!i 


60 
51 


007044 
9.008278 


2058 


997745 
9.997732 


21 
21 


009298 


2080 


990702 


10 
9 


2052 


9.010546 


2074 


10.989454 


62 


009510 


2046 


997719 


21 


011790 


2068 


988210 


8 


53 


010737 


2040 


997708 


21 


013031 


2062 


986989 


7 


54 


011962 


2034 


997693 


22 


014288 


2056 


985732 


6 


55 


013182 


2029 


997680 


22 


016502 


2051 


984498 


6 


56 


' 014400 


2023 


997687 


22 


010732 


2045 


983268 


4 


57 


015613 


2017 


997654 


22 


017959 


2040 


982041 


3 


58 


016824 


2012 


907641 


22 


019183 


2033 


980817 


2 


59 


018031 


2006 


997628 


22 


020403 


2028 


979597 


1 


60 


019235 


2000 


997614 


22 


021620 


2023 


978380 







Cosine 1 


1 


Sine 1 1 


Cotang. 




'I'Bor 








17 




d4i)6 


greet. 







24 


(6 Degreesi) a 


TABLE OF LOGASITBMIO 






Bine 


1 D. 


1 Cosine | D. 


1 Tang. 


D. 


Cbtang. 1 • 1 





9.019236 


2000 


9.997614 


22 


9.021620 


2023 


10.978380 


US 


1 


020435 


1995 


997601 


22 


022834 


2017 


977166 


69 


3 


021632 


1989 


997588 


22 


024044 


2011 


976956 


58 


3 


022826 


1984 


997574 


22 


025251 


2006 


974749 


67 


4 


024016 


1978 


997561 


22 


026455 


2000 


973645 


66 


5 


026203 


1973 


997547 


22 


U27665 


1995 


972345 


55 


6 


026386 


1967 


997534 


23 


028852 


1990 


971148 


54 


7 


027667 


1962 


997620 


23 


030046 


1965 


969954 


53 


8 


028744 


1967 


997507 


23 


031237 


1979 


968763 


62 


9 


029918 


1951 


997493 


23 


032425 


1974 


967576 


51 


10 
11 


031089 


1947 


997480 


23 
23 


033609 


1969 


966391 


50 
49 


9.032267 


1941 


9.997466 


9.034791 


1964 


10.905209 


12 


033421 


1936 


997452 


23 


035969 


1958 


964031 


48 


13 


034682 


1930 


997439 


23 


037144 


1953 


962856 


47 


14 


036741 


1925 


997425 


23 


038316 


1948 


961684 


46 


16 


036896 


1920 


997411 


23 


039485 


1943 


960515 


45 


16 


038048 


1915 


997397 


23 


040651 


1938 


969349 


44 


17 


039197 


1910 


997383 


23 


041813 


1933 


958187 


43 


18 


040342 


1905 


997369 


23 


042973 


1928 


957027 


42 


19 


041485 


1899 


997355 


23 


044130 


1923 


95.5870 


41 


20 

21 


042625 


1894 


997341 


23 
24 


045284 


1918 


964716 


40 
39 


9.043762 


1889 


9.997327 


9.046434 


1913 


10.953666 


22 


044895 


1884 


997313 


24 


047582 


1908 


962418 


38 


23 


046026 


1879 


997299 


24 


048727 


1903 


951273 


^ 


24 


047154 


1875 


997285 


24 


049869 


1898 


960131 


36 


26 


048279 


1870 


997271 


24 


051008 


1893 


948992 


35 


26 


049400 


1865 


997257 


24 


052144 


1889 


947866 


34 


27 


050619 


1860 


997242 


24 


053277 


1884 


946723 


33 


28 


061635 


1855 


997228 


24 


054407 


1879 


946693 


32 


29 


062749 


1850 


997214 


24 


065635 


1874 


944465 


31 


30 
31 


053869 
054966 


1845 


997199 
9.997185 


24 
24 


056669 
9.057781 


1870 


943341 


30 
29 


1841 


1865 


10.942219 


32 


060071 


1836 


997170 


24 


058900 


1869 


941100 


28 


33 


067172 


1831 


997156 


24 


060016 


1855 


939984 


37 


34 


058271 


1827 


997141 


24 


061130 


1851 


938870 


26 


36 


059307 


1822 


997127 


24 


062240 


1846 


937760 


25 


36 


0G0460 


1817 


997112 


24 


063348 


1842 


936652 


24 


37 


061661 


1813 


997098 


24 


064453 


1837 


936647 


23 


3S 


002639 


1808 


997083 


26 


065566 


1833 


934444 


22 


39 


063724 


1804 


997068 


25 


066665 


1828 


933345 


21 


40 
41 


064806 
9.065886 


1799 


997063 


25 
26 


067763 


1824 


932248 


20 
19 


1794 


9.997039 


9.068846 


1819 


10.9311.64 


42 


066962 


■ 1790 


997024 


25 


069938 


1815 


930062 


18 


43 


068036 


1786 


997009 


25 


071027 


1810 


928973 


17 


44 


009107 


1781 


996994 


25 


072113 


1806 


927887 


16 


46 


070176, 


1777 


996979 


25 


073197 


1802 


926803 


15 


46 


071242i 


1772 


996964 


25 


074278 


1797 


925722 


14 


47 


072306; 


1768 


996949 


25 


075366 


1793 


924644 


13 


48 


073366 


1763 


996934 


25 


076432 


1789 


923568 


12 


49 


074424 


1769 


996919 


25 


077605 


1784 


922495 


11 


60 
61 


076480 


1765 


996904 


25 
25 


078676 
9.079644 


1780 
1776 


921424 


10 
9 


9.076633 


1750 


9.996889 


10.020356 


62 


077583 


1746 


996874 


25 


080710 


1772 


919290 


8 


63 


078031 


1742 


996858 


25 


081773 


1767 


918227 


7 


64 


079676 


1738 


996843 


25 


082833 


1763 


917167 


6 


65 


080719 


1733 


996828 


25 


083891 


1769 


916109 


5 


66 


OS 1759 


1729 


996812 


26 


084947 


1755 


915053 


4 


67 


082791 


1725 


996797 


26 


086000 


1751 


914000 


3 


68 


083832 


1721 


996782 


26 


087050 


1747 


912960 


2 


69 


084864 


1717 


996766 


26 


088098 


1743 


911902 


1 


60 


0868941 


1713 


996761 


26 


089144 


1738 


910856 





_l 


Cosine 




Bine | 


(Joiaiii;. 




Tang. M. | 



83 Degroei. 



8BIX8 AND TANOEim. (7 Degrees.) 



26 



M. 


Sine 


D. 


Cocine | D. 


Tanij. 


D. 1 


Cotani;. ( 





0.085894 


1713 


9.996761 26 


9.089144 


1738 


10.910856 


60 


1 


086922 


1709 


996735 26 


090187 


1734 


909813 


69 


s 


087947 


1704 


996720 26 


091228 


1730 


908772 


68 


3 


088970 


1700 


996704 


26 


092266 


1727 


907734 


67 


4 


089990 


1696 


996688 


26 


093302 


1722 


906698 


56 


5 


091008 


1692 


996673 26 


094330 


1719 


905664 


55 


6 


092024 


1688 


996657 26 


, 095367 


1716 


904603 54 


7 


093037 


1684 


996641 26 


096395 


1711 


903605 63 


8 


094047 


1680 


996626 


26 


097422 


1707 


902578 


52 


9 


095056 


1676 


996610 


26 


098446 


1703 


901.554 


51 


10 
11 


096062 


1673 


996594 


26 
27 


099468 


1699 


900532 


60 


9.097066 


1668 


9.996578 


9.100487 


1696 


10.899513 


49 


12 


098066 


1665 


9965C2 


27 


101504 


1691 


898496 


48 


13 


099065 


1661 


996.546 


27 


102619 


1687 


897481 


47 


14 


100062 


1657 


996630 


27 


103532 


1684 


896468 


46 


15 


101056 


1653 


996514 


27 


104542 


1680 


895468 


45 


16 


102048 


1649 


996498 


27 


105560 


1676 


894460 


44 


17 


103037 


1645 


996482 


27 


106666 


1672 


893444 


43 


18 


104025 


1641 


996465 


27 


107559 


1669 


892441 


42 


19 


10.5010 


16.38 


996449 


27 


108660 


1665 


891440 


41 


20 
21 


105992 


1634 


996433 
9.996417 


27 

27 


109559 


1661 


890441 


40 


9.106973 


1630 


9.110556 


1658 


10.889444 


39 


22 


107951 


1627 


996400 


27 


111551 


1654 


888449 


38 


23 


108927 


1633 


996384 


27 


112543 


1650 


887457 


37 


24 


109901 


1619 


996368 


27 


113533 


1646 


886467 


36 


25 


110873 


1616 


996351 


27 


114521 


1643 


885479 


35 


26 


111842 


1612 


996335 


27 


115507 


16.39 


884493 


34 


27 


112809 


1608 
1605 


996318 


27 


116491 


1636 


883509 


33 


28 


113774 


996302 


28 


117472 


1632 


882628 


32 


29 


114737 


1601 


996285 


28 


118452 


1629 


881548 


31 


80 
31 


115698 


1597 


996269 


28 
28 


1 19429 


1626 


880571 
10.879596 


30 


9.116656 


1594 


9.996252 


9.120404 


1622 


29 


32 


117613 


1590 


996235 


28 


121377 


1618 


878623 


28 


33 


118567 


1587 


996219 


28 


122348 


1616 


877652 


27 


34 


119519 


1583 


996202 


28 


123317 


1611 


876683 


26 


35 


120469 


1580 


996185 


28 


124284 


1607 


876716 


25 


36 


121417 


1576 


996168 


28 


125249 


1604 


874751 


24 


37 


122362 


1673 


996151 


28 


126211 


1601 


873789 


23 


38 


123306 


1569 


996134 


28 


127172 


1597 


672828 


22 


39 


124248 


1566 


996117 


28 


128130 


1594 


871870 


21 


40 
41 


125187 


1562 


996100 


28 
29 


129087 


1691 


870913 


20 


9.126125 


1559 


9.996083 


9.130041 


1587 


10.869969 


19 


42 


127060 


1656 


996066 


29 


130994 


1684 


869006 


18 


43 


127993 


1552 


996049 


29 


131944 


1681 


868056 


17 


44 


128925 


1549 


9960.32 


29 


132893 


1877 


867107 


16 


45 


12985'1 


1645 


996016 


29 


133839 


1574 


866161 


16 


46 


130781 


1542 


995998 


29 


134784 


1671 


866216 


14 


47 


131706 


1539 


995980 


29 


135726 


1567 


864274 


13 


48 


132630 


1535 


995963 


29 


136667 


1664 


863333 


12 


40 


133551 


1532 


995946 


29 


137606 


1661 


862396 


11 


50 
51 


134470 


1529 


995928 


29 
29 


138.542 


1568 


861468 


10 


9.135387 


1526 


9.995911 


9.139476 


1665 


10.860624 


9 


52 


136303 


1622 


995894 


29 


140409 


1661 


869691 


8 


53 


137216 


1519 


996876 


29 


141340 


1648 


858660 


7 


54 


138128 


1616 


996859 


29 


142269 


1545 


867731 


6 


55 


139037 


1612 


996841 


29 


143196 


1642 


866804 


5 


56 


139944 


1609 


996823 


29 


144121 


1639 


866879 


4 


57 


140850 


1606 


996806 


29 


145044 


1636 


8549.56 


3 


58 


141764 


1603 


995788 


29 


146966 


1532 


864034 


2 


50 


142656 


1600 


996771 


29 


146886 


1629 


863116 


1 


60 


1435.55 


1496 


995753 29 


147803 


1626 


852197 





_l 


Cosine 1 




bine 1 


1 folmifc-. 1 


1 


Tuiif: 


pS- 



8K Degreci. 





26 


(8 Dogrees.; a table or LooAXiTBinc 






TT 


1 Sine 


1 D. 


1 Cosine | D. 


Twig. 


I D. 


Coiong. 1 1 







9.143555 


1496 


9.995753 


30 


: 9.147803 


1526 


10.8521971 601 




1 


144453 


1493 


995735 


30 


148718 


1628 


651282 


69 




2 


145349 


1490 


995717 


30 


149632 


1520 


860368 


58 




3 


. 146243 


1487 


995699 


30 


150544 


1517 


849466 


67 




4 


147136 


1484 


995681 


30 


151454 


1514 


848546 


56 




5 


148026 


1481 


995664 


30 


152363 


1511 


847637 


66 




G 


148915 


1478 


995646 


30 


153269 


1508 


846731 


64 




7 


149802 


1475 


995628 


30 


154174 


1505 


845826 


63 




8 


150686 


1472 


995610 


30 


155077 


1502 


844923 


52 




9 


151569 


1469 


995591 


30 


155978 


1499 


844022 


21 




10 
11 


152451 


1466 


995573 


30 
30 


156877 


1496 


843123 


60 
49 




9 153330 


1463 


9.995555 


9.157775 


1493 


10.842225 




12 


154208 


1460 


995537 


30 


168671 


1490 


841329 


48 




13 


155083 


1457 


995519 


30 


159565 


1487 


840435 


47 




14 


155957 


1454 


995501 


31 


160457 


1484 


839543 


46 




15 


156830 


1451 


995482 


31 


161347 


1481 


838653 


46 




16 


157700 


1448 


995464 


31 


162236 


1479 


837764 


44 




17 


158569 


1445 


995446 


31 


163123 


1476 


836877 


43 




18 


159435 


1442 


995427 


31 


164008 


1473 


835992 


48 




19 


160301 


1439 


995409 


31 


164892 


1470 


835108 


41 




20 
21 


161164 


1436 


995390 


31 
31 


165774 
9.166654 


1467 


834226 


40 
39 




9.162025 


1433 


9.995372 


1464 


10.833346 




22 


162885 


1430 


995353 


31 


167532 


1461 


832468 


38 




23 


163743 


1427 


995334 


31 


168409 


1458 


831591 


37 




24 


164600 


1424 


995316 


31 


169284 


1455 


830716 


36 




25 


165454 


1422 


995297 


31 


170157 


1453 


829843 


35 




26 


166307 


1419 


995278 


31 


171029 


1460 


828971 


34 




27 


167159 


1416 


995260 


31 


171899 


1447 


828101 


33 




28 


168008 


1413 


995241 


32 


172767 


1444 


827233 


32 




29 


168856 


1410 


995222 


32 


173634 


1442 


826366 


31 




30 
31 


169702 


1407 


995203 


32 
32 


174499 


1439 


825501 


30 
29 




9.170547 


1405 


9.995184 


9.175362 


1436 


10.824638 




32 


171389 


1402 


995165 


32 


176224 


1433 


823776 


28 




33 


172230 


1399 


995146 


32 


177084 


1431 


822916 


27 




34 


173070 


1396 


995127 


32 


177942 


1428 


822058 


26 




35 


173908 


1394 


995108 


32 


178799 


1426 


821201 


26 




36 


174744 


1391 


995089 


32 


1796.55 


1423 


820346 


24 




37 


176578 


1388 


995070 


32 


180508 


1420 


819492 


23 




38 


176411 


1386 


995051 


32 


181360 


1417 


818640 


22 




39 


177242 


1383 


995032 


32 


182211 


1415 


817789 


21 




40 
41 


178072 


1380 


995013 


32 
32 


183059 


1412 


816941 


20 
19 




9.178900 


1377 


9.994993 


9.183907 


1409 


10.816093 




42 


179726 


1374 


994974 


32 


184752 


1407 


815248 


18 




43 


180551 


1372 


994955 


32 


185597 


1404 


814403 


17 




44 


181374 


1369 


994935 


32 


186439 


1402 


813561 


16 




45 


182196 


1366 


994916 


33 


187280 


1399 


812720 


15 




46 


183016 


1364 


994896 


33 


188120 


1396 


811880 


14 




47 


1838:W 


1361 


994877 


33 


188958 


1393 


811042! 131 




48 


184651 


1359 


994857 


33 


189794 


1391 


810206 


12 




49 


185466 


1350 


994838 


33 


190629 


1389 


809371 


U 




60 
51 


186280 


1353 
1351 


994818 


33 
33 


191462 


138G 


808638 
10.807706 


10 
9 




9.187092 


9.994798 


9.192294 


1384 




52 


187903 


1348 


994779 


33 


193124 


1381 


80C876 


8 




53 


188712 


1346 


994759 


33 


1939.53 


1379 


806047 


7 




54 


189519 


1343 


994739 


33 


194780 


1376 


806220 


6 




55 


190325 


1341 


994719 


33 


195606 


1374 


804394 


6 




56 


191130 


1338 


994700 


33 


196430 


1371 


803570 


4 




57 


191933 


1336 


994680 


33 


197253 


1369 


802747 


3 




58 


192734 


1333 


994660 


33 


198074 


1366 


801926 


2 




59 


193534 


1330 


994640 


33 


198894 


1364 


801106 


7 




60 


194.332 


1328 


994620 


33 


199713 


1361 


800287 









Cosine 




Sine j 


Cotaiig. 




Taiu;. 1 M. 1 



81 Degrees. 





snos AVD TA^4lF^T9. {0 rc^pees.^ 


17 


M. 


Sine 1 


Dl 


Carrie 1 D 


'. T«ri. 


! P. 


; rorxn*. 1 





9.19433-^ 


1328 


9.994620 33 


S. 199713 
200329 


1 1361 


10.8UO287. 60 


1 


1951£9 


1326 


994600 33 


1359 


7i*947ll59 


2 


195925 


13S3 


994580 33 


20i;S^15 


135S 


798655 58 


3 


196719 


1321 


S94560 34 


202159 


13»4 


797841 57 


4 


197511 


1318 


994540 M 


202971 


1352 


797029 56 


5 


198302 


1316 


994519 31 


2*}37S2 


1.349 


79€2!8 55 


6 


199091 


1313 


994499 34 


204592 


1347 


705408 M 


7 


199S79 


1311 


994479 34 


205400 


I 1^43 


794600 53 


8 


200666 


1308 


994459 34 


206207 


. 1»42 


793793 52 


9 


201451 


1306 


i 99443S 34 


207013 


l»4i3 


792987 51 


10 
11 


202234 


1304 
I3«31 


! 994418 34 
9.994397 34 


207SI7 

9.208619 


1338 


792183 50 
10.7913S1 49 


9.203017 


1335 


12 


203797 


1299 


994377 34 


20W20 


1333 


790580 48 


13 


204577 


1296 


994357 34 


210220 


1331 


789780 47 


14 


205354 


1294 


994336 34 


2110IS 


1328 


788982-46 


16 


206131 


1292 


994316 34 


211S15 


1326 


788185; 45 


16 


206906 


1289 


994295 .34 


212611 


1324 


7S7«S9; 44 


17 


207679 


1287 


994274; 35 


213405 


1321 


786595. 43 


18 


208452 


12S5 


994254 35 


214198 


1319 


785802' 42 


19 


209222 


1282 


994233 35 


214989 


1317 


7850 ir 41 


20 
21 


209992 


1280 


994212 35 
9.994191.35 


215780 

9.2I6568' 


1315 


7^4220 40 


9.21U760 


127S 


1312 


10.7'?3432 


391 


22 


21 1520 


1275 


994171.35 


217356 


I31U 


7S2644 3S 


23 


212291 


1273 


994150; 35 


218142 


1308 


781858 37 


24 


213C55 


1271 


99-1129' 35 


21S926 


1:305 


781074 36 


26 


213*518 


1268 


99410^ 35 


219710, 


1303 


7802U0 35 


26 


214579 


1266 


99 405?; 35 


220492. 


1301 


7795«)8 34 


27 


215338 


1264 


9940*56 35 


221272 


1299 


778728:33 


28 


216097 


1261 


994ai5i 35 


222052 


1297 


777948' 32 


29 


216do4 


1259 


994024 b5 


222'; 30 


1294 


777170131 


30 
31 


217600 
9.218363 


1C57 
1255 


994003 


351 

35! 


22360o 
9.224382 


1292 


776.394! 30 


9.993981; 


1290 


10.775613! 


29 1 


32 


219116 


1253 


993960 35 


225156 


1288 


7748-U; 28 


33 


219S6S 


1250 


993939! 35 


225929 


1286 


774071 27 


34 


220613 


1248 


9939 IS .35 


226700 


1284 


773300 26 


35 


221367 


1246 


993890 


36 


227471 


1281 


772529! 25 


36 


222115 


1244 


99:5875 


36 


228239 


1279 


771761 


24 


37 


222861 


1242 


993854 


36 


229007 


1277 


770993 


23 


38 


223606 


1239 


993832 


36 


229773 


1275 


770227 


22 


39 


224^49 


1237 


993811 


36 


230539 


1273 


769461 


21 


40 

41 


22509--i 
9.225333 


1235 


993789 
9.99376S 


36 
36 


231302 


1271 


768698 
10.767935 


20 
19 


1238 


9.2320G5 


1269 


42 


226573 


1231 


99374G 


36 


232826 


1267 


767174 


18 


43 


227211 


1228 


993725 


36 


2335SG 


1265 


766414 


17 


44 


228048 


122C 


993703: 36 


2;J4315 


12fi2 


765655 


16 


45 


228784 


1224 


993G81 3G 


235103 


1260 


764897 


15 


46 


229518 


122C 


99:i660 


3G 


i:358o9 


1258 


764141 


14 


47 


230252 


1220 


993638 


36 


23GHM 


1256 


763386 


13 


48 


230994 


1218 


993016 


36 


237368 


1254 


762632 


12 


49 


231714 


1216 


993594 


37 


238120 


1252 


701880 


11 


50 
51 


202444 


1214 


993572 


37 
37 


238872 


1250 


701128 
10.760378 


10 
9 


9.233172 


1212 


9.993550 


9.239622 


1248 


52 


233899 


1209 


993528 


37 


240371 


1246 


759620 


8 


53 


234625 


1207 


993506 


37 


241118 


1244 


758882 


7 


54 


235349 


1205 


993484 


37 


241865 


1242 


758135 


6 


55 


236073 


1203 


993402 


37 


242610 


1240 


757390 


5 


56 


236795 


1201 


993440 


37 


243354 


1238 


756G16 


4 


57 


237515 


1190 


993418 


37 


244097 


1236 


755903 


3 


58 


238235 


1197 


99339G 


37 


244839 


1234 


755161 


2 


59 


238953 


1195 


993374 


37 


245579 


1232 


76442' ^ ■ ■ 


60 


239670 


1193 


993351 


37 


246319 


1230 


7536 





d 



CuKinb 



17* 



time I I Coiaiig. I 
8U Degree*. 



'J'BIK 



££ 



28 



(10 Degrees.) a tacle op LooABinnac 



M.| 


SiiiH 


D. 1 


C<»sine | D. 


1 'I'anff. 


D. 


Cotang. 1 1 





9.239670 


1193 


9.993351 


37 


9.246319 


1230 


10.753681 


60 


1 


240386 


1191 


993329 


37 


247057 


1228 


752943 


59 


2 


241101 


1189 


993307 


37 


247794 


1226 


752206 


58 


3 


241814 


1187 


993285 


37 


248630 


1224 


761470 


57 


4 


242526 


1185 


993262 


37 


249264 


1222 


750736 


56 


5 


243237 


1183 


993240 


37 


249998 


1220 


760002 


56 


6 


243947 


1181 


993217 


38 


260730 


1218 


749270 


54 


7 


244656 


1179 


993195 


38 


251461 


1217 


748539 


63 


8 


245363 


1177 


993172 


38 


252191 


1215 


747809 


62 


9 


246069 


1175 


993149 


38 


252920 


1213 


747080 


51 


10 
11 


246775 
9.247478 


1173 
1171 


993127 


38 
38 


253648 


1211 


746352 


50 
49 


9.993104 


9.254374 


1209 


10.745626 


12 


248181 


1169 


993081 


38 


2.55100 


1207 


744900 


48 


13 


248883 


1.67 


993059 


38 


255824 


1205 


744 176 


47 


14 


249583 


1165 


993036 


38 


256.547 


1203 


743453 


46 


15 


250282 


1163 


993013 


38 


257269 


1201 


742731 


45 


IB 


250980 


1161 


992990 


38 


257990 


1200 


742010 


44 


17 


251677 


1159 


992967 


38 


258710 


1198 


741290 


4S 


18 


252373 


1158 


992944 


38 


259429 


1196 


740571 


42 


19 


253067 


1156 


992921 


38 


260146 


1194 


739854 


41 


20 
21 


253761 


1154 


992898 
9.992875 


38 
38 


260863 


1192 


739137 


40 
39 


9.264453 


11,52 


9.261578 


1190 


10.738422 


22 


255144 


1150 


992852 


38 


262292 


1189 


737708 


38 


23 


t:55834 


1148 


992829 


39 


263005 


1187 


736995 


37 


24 


l;r'Cr)23 


1146 


• 9928U6 


39 


263717 


1185 


736283 


36 


25 


257211 


1144 


982783 


39 


264428 


1183 


735572 


35 


26 


257898 


1142 


992759 


.^9 


265138 


1181 


734862 


34 


27 


258583 


1141 


992730 


39 


265847 


1179 


7341.53 


33 


28 


2592G8 


1139 


992713 


39 


'^66555 


1178 


733^145 


32 


29 


259951 


1137 


992G90 


39 


267261 


1176 


732739 


31 


30 
31 


260633 
9.261314 


J 135 


992666 
9.992643 


39 
39 


267967 


1174 
1172 


732033 
10.731329 


30 
29 


1133 


9.268671 


32 


261994 


1131 


992619 


39 


269375 


1170 


730625 


28 


33 


262673 


1130 


99259G 


39 


270077 


1169 


729923 


27 


34 


263551 


1128 


992572 


39 


270779 


1167 


729221 


26 


35 


264027 


1126 


992549 


39 


271479 


1165 


728521 


25 


36 


26471)3 


1124 


092525 


39 


272178 


1164 


727822 


24 


37 


265377 


1122 


992501 


39 


272876 


1162 


. 727124 


23 


38 


266051 


1120 


992478 


40 


273573 


1160 


726427 


22 


30 


206723 


1119 


992454 


40 


274269 


1158 


72573 t 


21 


40 
41 


2G7305 
9.26^065 


1117 


992430 
9.992400 


40 
40 


274964 


1157 


725036 


20 
19 


1115 


9.275658 


1155 


10.724342 


42 


268734 


1113 


9923S2 


40 


276351 


1153 


723649 


18 


43 


269402 


nil 


992359 


40 


277013 


1151 


722937 


17 


4'i 


27U0C9 


1110 


992335 


40 


277734 


1150 


722266 


IS 


15 


270735 


1108 


992311 


40 


278424 


1148 


721576 


15 


46 


271400 


1106 


992287 


40 


279113 


1147 


720887 


14 


47 


272064 


1105 


992263 


40 


279801 


1145 


720199 


13 


48 


272726 


1103 


992239 


40 


280488 


1143 


719512 


13 


49 


273388 


1101 


992214 


40 


281174 


1141 


718826 


11 


50 

51 


274049 


1099 


992190 


40 
40 


281858 


1140 
1138 


718142 


10 
9 


9.274708 


1098 


9.992100 


9.282542 


10.717458 


52 


275367 


1096 


992142 


40 


283225 


1136 


716775 


8 


63 


276024 


1094 


992117 


41 


283907 


1135 


716093 


7 


54 


276681 


1092 


992093 


41 


284588 


1133 


715412 


6 


65 


277337 


1091 


992069 


41 


285268 


1131 


714732 


6 


56 


277991 


1089 


992044 


41 


285947 


1130 


714053 


4 


57 


278644 


1087 


992020 


41 


286624 


1128 


713376 


3 


58 


279297 


1086 


991996 


41 


287301 


1126 


712699 


2 


69 


279948 


1084 


991971 


41 


287977 


1125 


712(123 


1 


60 


280590 


1082 


991947 


41 


2S8652 


1123 


711348 







Ctwiiie 1 

• 




Sine 1 


Cl)lall^. 1 


1 


lAix^. 1 M. 1 



07 Degrees. 



mnti A^TD TAiiGEirTs. (11 Degrees.) 



M. 


1 Sine 


1 n.' 


1 Cn«in« D. 


1 Tanr. 


1 D. 


I (iNAnff. 1 ^ 


"o" 


9. 280599 


1082 


9.991947 


41 


9.2886.52 


1123 


10.711348 


60 


1 


281248 


1081 


991922 


41 


289326 


1122 


710674 


59 


2 


231897 


1079 


991897 


41 


289999 


1120 


710001 


58 


3 


232544 


1077 


991873 


41 


290671 


1118 


709329 


57 


4 


233190 


1076 


991848 


41 


291342 


1117 


708658 


56 


5 


283S36 


1074 


991823 


41 


292013 


1115 


707937 


56 


6 


284480 


1072 


991799 


41 


292682 


1114 


707318 


64 


7 


285124 


1071 


991774 


42 


293350 


1112 


706650 


63 


8 


286766 


1069 


991749 


42 


294017 


nil 


705983 


62 


9 


286408 


1067 


991724 


42 


294684 


1109 


706316 


61 


10 
11 


287048 
9.287687 


1066 


991699 
9.991674 


42 
42 


295349 
9.296013 


1107 


704651 


60 

49 


1064 


1106 


10.703987 


12 


288326 


1063 


991649 


42 


296677 


1104 


703323 


48 


13 


288964 


1061 


991624 


42 


297339 


1103 


702661 


47 


14 


239600 


1059 


9*1599 


42 


29S001 


1101 


701999 


46 


15 


290236 


1058 


991574 


42 


2J8662 


1100 


701338 


46 


16 


290870 


1056 


991.549 


42 


299322 


1098 


700678 


44 


17 


291504 


1054 


991524 


42 


2999S0 


1096 


700020 


43 


Id 


292137 


1053 


991498 


42 


300633 


1095 


699362 


42 


19 


292768 


1051 


991473 


42 


301295 


1093 


693705 


41 


20 
21 


293399 
9.2i)l!)iy 


1050 


9.M44S 


42 
42 


3^1951 
9.30^(iJV 


1002 
lOilO 


698049 


40 
39 


10^^ 


10.697393 


22 


294658 


1046 


991397 


42 


30.V26 1 


1089 


696739 


38 


23 


295286 


1045 


991372 


43 


30.3914 


1087 


696086 


37 


24 


295913 


1043 


9913^6 


43 


.S04567 


1086 


695433 


36 


25 


296539 


1042 


991321 


43 


305218 


1084 


691782 


35 


26 


297164 


1040 


991295 


43 


305369 


1083 


694131 


34 


27 


297788 


1039 


991270 


43 


306519 


1081 


693481 


33 


28 


298412 


1037 


991244 


43 


307168 


1080 


692832 


32 


2J 


299034 


1036 


991218 


43 


307815 


1078 


692185 


31 


30 
31 


299655 
9.300276 


1034 


991193 


43 
43 


308463 
9.309109 


1077 


691537 


30 
29 


1032 


9.991167 


1075 


10.690891 


32 


300895 


1031 


991141 


43 


309754 


1074 


690246 


28 


33 


301514 


1029 


991115 


43 


310398 


1073 


639602 


27 


34 


302132 


1028 


991090 


43 


811042 


1071 


688958 


26 


35 


302748 


1026 


991064 


43 


311685 


1070 


688315 


26 


36 


303364 


1025 


991038 


43 


312327 


1068 


687673 


24 


37 


303^79 


1023 


991012 


43 


312967 


1067 


687033 


23 


38 


334593 


1022 


9909S6 


43 


31300S 


1005 


6S6392 


22 


39 


305207 


1020 


990960 


43 


314247 


1064 


685753 


21 


40 
41 


305319 
9.306430 


1019 


990934 44 
9.990908 44 


314885 
9.315523 


1062 


685115 


20 
19 


1017 


1061 


10.684477 


42 


307041 


1016 


990882 


44 


316159 


1060 


633841 


18 


43 


3073-.0 


1014 


990855 


44 


316795 


1058 


683205 


17 


44 


308259 


1013 


990829 


44 


317430 


1057 


682570 


16 


45 


308867 


1011 


990803 


44 


318064 


1056 


681936 


15 


46 


309474 


1010 


990777 


44 


318697 


1054 


681.303 


14 


47 


310080 


1008 


990750 


44 


319329 


1053 


680671 


13 


48 


310685 


1007 


990724 


44 


319961 


1061 


680039 


12 


49 


311289 


1005 


990697 


44 


320592 


1050 


679408 


11 


50 
51 


311893 
9.312495 


1004 


990671 


44 


321222 
9.321851 


1048 


678778 


10 
9 


1003 


9.990644 


44 


1047 


10.678149 


52 


313097 


1001 


990618 44 


322479 


1046 


677621 


8 


53 


313698 


1000 


990591 44 


323106 


1044 


676894 


7 


54 


314297 


998 


990565 44 


323733 


1043 


676267 


6 


55 


314897 


997 


990538 44 


324358 


1041 


676642 


5 


56 


315495 


996 


990511 45 


324983 


1040 


676017 


4 


57 


316092 


994 


«)90485 45 


325607 


1039 


674393 8 


58 


316689 


993 


990458 45 


326231 


1037 


673769 2 


59 


317284 


991 


990431 45 


326853 


1036 


6731471 1 


60 


317879 


990 


990404 45 


327476 


1035 


672526 




1 Gnrine | 




Suie 1 


1 Ck>taiirf. 


. 


1 Tai«. 


L ■ 






?R 



do 


(12 Degrees.) a 


TABLE OF XOOARTTHBUC 




"m" 


Sine 


1 I>. 


1 Cofine 1 D. 


1 Tang. 


1 D. 


1 Cofaiip. i 





9.317879 


990 


9.990404 


45 


9.327474 


1035 


10.672526 


60 


1 


318473 


988 


990378 


45 


328095 


1033 


671905 


59 


2 


319066 


987 


990351 


45 


328715 


1032 


671285 


58 


3 


319658 


986 


990.324 


45 


329334 


1030 


670666 


57 


4 


320249 


984 


990297 


45 


329953 


1029 


670047 


56 


5 


320840 


983 


990270 


45 


330570 


1028 


669430 


56 


6 


321430 


982 


990243 


45 


331187 


1026 


C68813 


54 


7 


322019 


980 


990215 


45 


331803 


1025 


668197 


53 


8 


322607 


979 


990188 


45 


332418 


1024 


667582 


52 


9 


323194 


977 


990161 


45 


333033 


1023 


666967 


51 


10 
11 


323780 
9.32^1366 


976 


990134 
9.990107 


45 
46 


333646 


1021 


666354 
10.665741 


50 


975 


9.334259 


1020 


49 


12 


324950 


973 


990079 


46 


334871 


1019 


665129 


48 


13 


325534 


972 


990052 


46 


335482 


1017 


664518 


47 


14 


326117 


970 


990025 


46 


336093 


10)6 


663907 


46 


15 


326700 


969 


98U997 


46. 


336702 


1015 


663298 


45 


16 


327281 


968 


989970- 


46 


337311 


1013 


662689 


44 


17 


327862 


966 


989942 


46 


337919 


1012 


662081 


43 


18 


328442 


965 


989915 


46 


338527 


1011 


661473 


42 


19 


329021 


964 


989887 


46 


339133 


1010 


660867 


41 


20 
21 


329599 


962 


9S9860 


46 
46 


339739 


1008 


660261 


40 


9.33017'? 


961 


9.989832 


9.340344 


1007 


10.659656 


39 


22 


330753 


960 


989804 


46 


340948 


1006 


659052 


38 


23 


331329 


958 


989777 


46 


341552 


1004 


658448 


37 


24 


331903 


957 


989749 


47 


342155 


1003 


657845 


36 


25 


332478 


956 


989721 


47 


342767 


1002 


657243 


35 


26 


333051 


954 


989693 


47 


343358 


1000 


656642 


34 


27 


333624 


953 


989665 


47 


343958 


999 


6.%042 


33 


28 


334195 


952 


989637 


47 


344568 


998 


655442 


32 


29 


334766 


950 


989609 


47 


345157 


997 


654843 


31 


30 
31 


335337 


949 


989582 


47 
47 


345755 
9.346353 


996 


654245 


30 


9.335906 


948 


9.989.553 


994 


10.653647 


29 


32 


336475 


946 


989525 


47 


346949 


993 


6.53051 


28 


33 


337043 


945 


989497 


47 


347545 


992 


652455 


27 


34 


337610 


944 


989169 


47 


348141 


991 


651859 


26 


35 


338176 


943 


989441 


47 


348735 


990 


6512G5 


25 


36 


3.?8742 


941 


989413 


47 


349329 


988 


650671 


24 


37 


339300 


940 


989384 


47 


349922 


987 


650078 


23 


38 


339871 


939 


989356 


47 


350514 


986 


649480 


22 


39 


340434 


937 


989328 


47 


351106 


985 


64P894 


21 


40 
41 


340996 


936 
935 


989300 


47 
47 


351697 


983 

982 


648303 


20 


9.341558 


9.989271 


9.352287 


10.647713 


19 


42 


342119 


934 


989243 


47 


352876 


981 


647124 


18 


43 


342679 


932 


989214 


47 


353465 


980 


646535 


17 


44 


343239 


931 


989186 


47 


354053 


979 


645947 


16 


45 


343797 


930 


989157 


47 


354640 


977 


645360 


15 


46 


344365 


929 


989128 


48 


355227 


976 


644773 


14 


47 


344912 


927 


989100 


48 


355813 


975 


644187 


13 


48 


345469 


926 


989071 


48 


356398 


974 


643602 


12 


49 


346024 


925 


989042 


48 


356982 


973 


643018 


11 


50 
51 


346579 


924 


989014 
9.988985 


48 
48 


357566 


971 
970 


642434 


10 


9.347134 


922 


9.358149 


10.641851 


9 


52 


347687 


921 


988956 


48 


358731 


969 


641269 


8 


53 


348240 


920 


988927 


48 


359313 


968 


640687 


7 


54 


348792 


919 


988898 


48 


359893 


967 


640107 


6 


55 


349343 


917 


988869 


48 


360474 


966 


639.526 


5 


56 


349893 


916 


988840 


48 


361053 


965 


638947 


4 


57 


350443 


915 


988811 


49 


361632 


963 


638368 


3 


58 


350992 


914 


988782 


49 


362210 


962 


637790 


2 


59 


351540 


913 


988753 


49 


r62787 


961 


637213 


1 


no 


352088 


911 


988724 


49 


363364 


960 


63063ri 






Co:iinc 



Sine 



I I Colang. I 
77 Dftgrees. 



I Tang I 



M. 





SINES AND TANGENTS. ^13 Degreefl 





31 


VL 


SiM« 1 


D. 


1 C»ine 1 1). 


1 'i'niii'. 


1 n. 


Co:an'j;. | 





9.352(W8 


911 


9.938724 


49 


9.303304 


960 


10.636630{60 


1 


352635 


910 


938695 


49 


333940 


959 


636060! 69 


2 


353181 


909 


983666 


49 


364515 


958 


635485 


is 


3 


353726 


908 


933630 


49 


365090 


957 


634010 


67 


4 


351271 


907 


988607 


49 


365604 


955 


6.34336! .56 


5 


354815 


905 


933573 


49 


360237 


954 


6i53763 


55 


6 


355358 


904 


933543 


49 


366310 


953 


6.33190 


54 


7 


355901 


903 


988519 


49 


367332 


952 


632618 


53 


8 


3504^13 


902 


938439 


49 


387953 


951 


632047 


52 


9 


356984 


901 


938460 


49 


368.524 


950 


631476; 51 


10 
11 


^57524 
9.3.)30;>4 


899 


938430 
9.938401 


49 
49 


369094 


949 


630906 


50 
49 


893 


9.369663 


•948 


10.630337 


12 


358003 


897 


988371 


49 


370232 


946 


(S29768 


48 


13 


359141 


896 


933342 49 


370799 


945 


620201 


47 


14 


359078 


895 


988312 ,50 


371307 


944 


628633 


46 


15 


3602 15 


893 


983282 


50 


371933 


943 


623067 


45 


16 


360752 


892 


983252 


50 


372499 


942 


627501 


44 


17 


361287 


891 


988223 


50 


373064 


941 


626936 


43 


18 


361822 


890 


938193 


50 


373629 


940 


626371 


42 


19 


302356 


889 


988163 


50 


374193 


939 


625307 


41 


20 
21 


362889 
9.3'>3i22 


888 


988133 
9.983103 


50 
50 


374756 
9.375319 


933 


625244 


40 
39 


887 


937 


10.621631 


22 


303054 


885 


933073 


50 


375831 


985 


624119 


38 


23 


304485 


884 


988043 


50 


376442 


934 


623553 


37 


24 


305010 


883 


983013 


59 


377003 


933 


622997 


36 


2.5 


365546 


882 


987933 


50 


377533 


932 


622437 


35 


26 


3S0075 


881 


987953 


50 


378122 


931 


621878 


34 


27 


3)6004 


880 


937922 


50 


378631 


930 


621319 


33 


28 


307131 


879 


987892 


50 


879239 


929 


620761 


32 


29 


307659 


87T 


937862 


50 


379797 


923 


620203 


31 


30 
31 


308185 
9.368711 


876 


987832 


51 
51 


380354 
9.380910 


927 


619646 


30 

29 


875 


9.937301 


926 


10.619000 


32 


309236 


874 


987771 


51 


331466 


925 


618534 


28 


33 


309761 


873 


987740 


51 


382020 


924 


617930 


27 


34 


370285 


872 


987710 


51 


332575 


923 


617425 


26 


35 


370^03 


871 


937679 


51 


383129 


922 


616871 


25 


35 


371330 


870 


937649 


51 


333682 


921 


616318 


24 


37 


371852 


869 


987618 


51 


384231 


920 


615766 


23 


33 


3r2:J73 


807 


987588 


51 


384780 


919 


61.5214 


22 


30 


372804 


806 


937557 


51 


38.')337 


918 


614663 


21 


40 
41 


373414 
9.373933 


805 


987526 


51 
51 


385338 
9.380438 


917 


614112 


20 
19 


864 


9.987496 


915 


10.613.562 


42 


374452 


863 


987465 


51 


336037 


914 


613013 


18 


43 


374970 


862 


937434 


51 


387536 


913 


612464 


17 


44 


3754S7 


861 


937403 


52 


388084 


912 


611916 


16 


45 


376003 


860 


987372 


52 


333631 


911 


611369 


16 


46 


.376519 


859 


987341 


52 


389178 


910 


610822 


14 


47 


377035 


858 


997310 


52 


389724 


909 


610276 


13 


48 


377549 


857 


987279 


52 


390270 


908 


609730 


12 


49 


378003 


856 


987248 


52 


390816 


907 


609185 


11 


60 


378577 


854 


987217 
9.987186 


52 
52 


391360 


906 


603640 


10 
9 


61 


9.379089 


853 


9.391903 


905 


10.603097 


62 


379601 


852 


987155 


52 


392447 


904 


607553 


8 


63 


.380113 


851 


987124 


52 


392989 


903 


607011 


7 


64 


380624 


850 


987092 


52 


393531 


902 


606469 


6 


66 


381134 


849 


987061 


52 


394073 


901 


605927 


6 


56 


381643 


848 


937030 


52 


394614 


900 


^05336 


4 


57 


382152 


847 


986998 


52 


395154 


899 


604«'«« 


3 


68 


3S2661 


846 


986967 


52 


395694 


898 


60^ -' 


69 


3831 68 


845 


986930 


1,2 


390233 


897 


P 


60 


383676 


844 


986904 


52 


396771 


896 


1 




_l 


C«)'*iiie 1 




Sine 1 


Cutaiit;. 










EE 


:* 


76 1 


[)egree8. 







32 


(14 Degrees.; a 


TABLE OF LOOABITHHIC 




M. 


Bine 


D. 


1 Cosine D. 


1 Tang. 


1 D. 


Cotang. 1 ] 





9.383676 


844 


9.986904 


62 


9.396771 


896 


10.6032-^9 


60 


1 


384182 


843 


986873 


63 


397309 


896 


602691 


59 


2 


384687 


842 


986841 


53 


397846 


895 


602154 


58 


3 


385192 


841 


986809 


53 


398383 


894 


601617 


67 


4 


885697 


840 


986778 


63 


398919 


893 


601081 


56 


5 


886201 


939 


986746 


53 


899455 


892 


600545 


56 


6 


386704 


838 


986714 


53 


399990 


891 


600010 


54 


7 


887207 


837 


986683 


53 


400524 


890 


699476 


58 


8 


387709 


836 


986661 


63 


401058 


889 


598942 


62 


9 


388210 


836 


986619 


63 


401591 


888 


698409 


61 


10 
11 


388711 


834 


986587 


53 
63 


402124 


887 


697876 


50 
49 


9.389211 


833> 


9.986655 


9.402656 


886 


10.697344 


12 


389711 


832 


986523 


63 


403187 


885 


596813 


48 


13 


390210 


831 


986491,63 


403718 


884 


596282 


47 


14 


390708 


830 


986459 


63 


404249 


883 


595751 


46 


15 


391206 


828 


986427 


53 


404778 


882 


695222 


46 


16 


391703 


827 


986396 


53 


405308 


881 


594692 


44 


17 


392199 


826 


986363 


64 


405836 


880 


694164 


43 


18 


392695 


826 


986331 


64 


406364 


879 


593636 


42 


19 


393191 


824 


986299 


54 


406892 


878 


693108 


41 


20 
21 


393685 


823 


986266 


54 
54 


407419 


877 


692581 


40 
39 


9.394179 


822 


9.986234 


9.407945 


876 


10.592055 


22 


394673 


821 


986202 


54 


408471 


876 


591529 


38 


23 


395166 


820 


986109 


54 


408997 


874 


691003 


87 


24 


395658 


819 


986137 


54 


409521 


874 


690479 


36 


25 


390150 


818 


986104 


54 


410045 


873 


589955 


36 


26 


396641 


817 


986072 


54 


410569 


872 


589431 


34 


27 


397132 


817 


986039 


54 


411092 


871 


688908 


33 


28 


397621 


816 


986007 


54 


411615 


870 


588385 


32 


29 


398111 


816 


985974 


54 


412137 


869 


587863 


31 


30 
31 


398600 


814 


985942 


64 
65 


412668 


868 


587342 


30 
29 


9.399088 


813 


9.985909 


9.413179 


867 


10.586821 


32 


399575 


812 


985876 


65 


413699 


866 


586301 


28 


33 


400f»62 


811 


985843 


65 


414219 


865 


685781 


27 


34 


400549 


810 


986811 


55 


414738 


864 


585262 


26 


35 


401035 


809 


986778 


55 


41.5257 


864 


584743 


25 


36 


401520 


808 


. 985745 


55 


415775 


863 


584225 


24 


37 


402005 


807 


985712 


55 


416293 


862 


583707 


23 


38 


402489 


806 


98567S 


55 


416810 


861 


.583190 


22 


39 


402972 


805 


985646 


55 


417326 


860 


582674 


21 


40 
41 


403455 
9.403938 


804 


985013 


55 
55 


417842 


859 


.582158 


20 
19 


803 


9.985580 


9.418358 


858 


10.581642 


42 


404420 


802 


985547 


55 


418873 


857 


.581127 


18 


43 


404901 


801 


985514 


55 


419387 


856 


580613 


17 


44 


405382 


800 


985480 


55 


419901 


855 


680099 


16 


45 


405862 


799 


985447 


55 


420415 


855 


579585 


15 


46 


406341 


798 


985414 


56 


420927 


854 


579073 


14 


47 


406820 


797 


985380 


56 


421440 


853 


678560 


13 


48 


407299 


796 


985347 


56 


421952 


852 


678048 


12 


49 


407777 


795 


985314 


56 


422463 


851 


577537 


11 


60 
61 


408254 


794 


985280 


56 
56 


422974 


850 


577026 


10 
9 


9.408731 


794 


9.985247 


9.423484 


849 


10..576516 


52 


409207 


793 


98.5213 


56 


423993 


848 


576007 


8 


53 


409682 


792 


985180 


56 


424503 


848 


675497 


7 


54 


4101571 


7pl 


985146 


50 


425011 


847 


574989 


6 


55 


410632 


790 


985113 


56 


425519 


846 


574481 


6 


56 


411106 


789 


985079 


56 


426027 


845 


573973 


4 


57 


411579 


788 


985045 


56 


426534 


844 


673466 


3 


58 


412052 


7S7 


985011 


56 


427041 


843 


672959 


M 


59 


412524 


786 


984978 


56 


427547 


843 


572453 


ll 


60 1 


412996 


785 


9S4944 


56 


428052 


842 


571948 


__l 


Coieiiie 1 


1 


dine | 


1 L'o.aiig. 




Tang 1 M. 



75 Degrees. 



• 


SDOBB AND TAiTonm. (15 Degrees.) 


S9 


~ 


Sine 


1 D. 


1 Cosine | D. 


1 Tan?. 


1 D. 


1 CiAans. 1 1 





9.412996 


785 


9.984944 


57 


9.428052 


842 


10.571948 


60 


1 


413467 


784 


984910 


67 


428557 


^841 


57144:3 


59 


2 


413938 


783 


984876 


57 


429062 


•840 


5709.38 


58 


3 


414408 


783 


984842 


67 


429566 


839 


570434 


57 


4 


414S78 


782 


984808 


57 


430070 


838 


569930 


56 


6 


416347 


781 


984774 


57 


430573 


838 


569427 


55 


6 


415815 


780 


984740 


57 


431075 


837 


568925 


54 


7 


416283 


779 


984706 


57 


431577 


836 


568423 


53 


8 


416751 


778 


984672 


57 


432079 


835 


567921 


52 


9 


417217 


777 


984637! 57 


432680 


834 


567420 


51 


10 

11 


417684 
9.418150 


770 
775 


984603 
9.984569 


57 
57 


4330^0 


833 


566920 


50 
49 


9.4.'?3580 


832 


10.666420 


12 


418615 


774 


984536 


67 


434080 


832 


5669V0 


48 


13 


419079 


773 


984.'>00; 57 


434579 


831 


56.5421 


47 


14 


419544 


773 


984466 


57 


435078 


830 


564922 


46 


15 


420007 


772 


984432 


58 


435576 


829 


564424 


45 


16 


420470 


771 


984397 


68 


4360P 


828 


563927 


44 


17 


420933 


770 


984363 


58 


436570 


828 


563430 


43 


18 


421395 


769 


984328 


58 


437067 


827 


562933 


42 


19 


421867 


768 


984294 


58 


437663 


826 


562437 


41 


20 
21 


422318 
9 422778 


767 


984269 


68 
68 


43S069 
9.4385,54 


825 


561941 


40 
39 


767 


9.984224 


824 


10.661446 


22 


423238 


766 


984190 


58 


439048 


823 


560962 


38 


23 


423G97 


765 


984165 


58 


439643 


823 


560467 


37 


24 


42H66 


764 


984120 


68 


440036 


822 


569964 


36 


25 


424615 


763 


984086 


68 


440629 


821 


5.59471 


35 


26 


425073 


762 


984060 


68 


4-11022 


820 


568978 


34 


27 


425530 


761 


984016 


58 


441514 


819 


558486 


33 


28 


426987 


760 


983981 


68 


442006 


819 


557994 


32 


29 


426443 


760 


983946 


58 


442497 


818 


567503 


31 


30 
31 


426899 


769 


083911 
9.983876 


68 
58 


442988 


817 


5.57012 
10.566521 


30 
29 


9.427354 


768 


9.443479 


816 


32 


427809 


767 


983840 


69 


443968 


816 


556032 


28 


33 


428263 


756 


983806 


69 


444458 


815 


915542 


27 


34 


428717 


765 


983770 


69 


444947 


814 


555063 


26 


35 


429170 


764 


983736 


69 


44.5436 


813 


554565 


25 


36 


429G23 


763 


983700 


59 


445923 


812 


5.54077 


24 


37 


430076 


762 


983664 


69 


446411 


812 


5.5.3689 


23 


38 


430527 


762 


983629 


69 


446898 


811 


553102 


22 


39 


430978 


751 


983694 


69 


447384 


810 


562616 


21 


40 

41 


431429 


760 


983668 


69 
69 


447870 


809 


562130 


20 
19 


9.431879 


749 


9.983623 


9.448366 


809 


10.661644 


42 


432329 


749 


983487 


69 


448841 


808 


551159 


18 


43 


432778 


748 


983462 


69 


449326 


807 


550674 


17 


44 


433226 


747 


983416 


59 


4498 10 


806 


560190 


16 


45 


433676 


746 


983381 


69 


460294 


806 


549700 


15 


46 


434122 


746 


983346 


69 


450777 


805 


549223 


14 


47 


434669 


744 


983309 69 


451260 


804 


548740 


13 


AS 


435016 


744 


983273 60 


451743 


803 


548267 


12 


49 


436462 


743 


9832.38 


60 


462226 


802 


547775 


11 


SO 
51 


436908 


742 


983202 


60 
60 


452706 


802 


547294 
10.546813 


10 
9 


9.436363 


741 


9.983166 


9.4.53187 


801 


52 


436798 


740 


983130 


60 


453668 


800 


546332 


8 


53 


437242 


740 


983094 


60 


464148 


799 


546862 


7 


54 


437686 


739 


9830.58 


60 


464628 


799 


546372 


6 


55 


438129 


738 


983022 


60 


466107 


798 


544893 


5 


56 


438572 


737 


982986 


60 


455686 


797 


514414 


4 


57 


439014 


736 


982960 


60 


466064 


796 


543936 


3 


58 


439456 


736 


982914 60 


456642 


796 


543468 


2 


59 


439897 


735 


982878 60 


467019 


795 


542981 


1 


60 


440338 


734 


982842160 


467496 


794 


542604 






Cimiiiii 1 




S^ine 


Coiang. 




I Ta.M?. 





74 



34 



(16 Degrees.) a tabls of looarithmic 



M. 


1 Bine 


D. 


1 Cosine D. 


1 Tang. 


1 D. 


1 Cotang. 1 1 


T 


9.440338 


734 


9.982842 


60 


9.457496 


794 


10.542504 


60 


1 


440778 


733 


982805 


60 


457978 


793 


642027 


69 


2 


441218 


732 


982769 


61 


458449 


793 


641551 


58 


3 


441658 


731 


982733 


61 


458926 


792 


541075 


67 


4 


442096 


731 


982696 


61 


459400 


791 


640600 


56 


6 


442535 


730 


982660 


61 


459875 


790 


640125 


56 


6 


442973 


729 


982624 


61 


460349 


790 


639651 


54 


7 


443410 


728 


982687 


61 


460323 


789 


639177 


53 


8 


443847 


727 


982551 


61 


461297 


788 


638703 


52 


9 


444284 


727 


982514 


61 


461770 


788 


638230 


51 


10 
11 


444720 


726 


982477 


61 
61 


462242 
9.462714 


787 


637758 
10.537286 


50 
49 


9.445155 


725 


9.982441 


786 


12 


445590 


724 


982404 


61 


463186 


785 


630814 


48 


13 


446025 


723 


982367 


61 


463658 


785 


536342 


47 


14 


446459 


723 


982331 


61 


464129 


784 


.535871 


46 


15 


446893 


722 


982294 


01 


464699 


783 


635401 


45 


16 


447326 


721 


982257 


61 


46506: 


783 


534;^:^1 


44 


17 


447759 


720 


982220 


62 


465539 


782 


634461 


43 


Id 


448191 


720 


982183 


62 


466008 


781 


6.S3992 


42 


19 


448623 


719 


982146 


62 


466476 


780 


533524 


41 


20 
21 


449054 


718 


982109 


62 
62 


466945 


780 


633055 
10.532587 


40 
39 


9.449485 


717 


9.982072 


9.467413 


779 


22 


449915 


716 


982035 


62 


467880 


778 


6.32120 


38 


23 


450346 


716 


981998 


62 


468347 


778 


631*653 


37 


24 


450775 


716 


981961 


62 


468814 


777 


631186 


36 


25 


451204 


714 


981924 


62 


469280 


776 


530720 


35 


26 


451632 


713 


981886 


62 


469746 


775 


630254 


34 


27 


452060 


713 


981849 


62 


470211 


775 


629789 


33 


28 


452488 


712 


981812 


62 


470676 


774 


629324 


32 


29 


452915 


711 


981774 


62 


471141 


773 


628859 


31 


30 
31 


453-342 
9.453768 


710 


981737 
9.981699 


62 
63 


471605 


7T3 


628395 
10.627932 


30 


710 


9.472068 


772 


32 


454194 


709 


981662 


63 


472532 


771 


627468 


:^ 


33 


454619 


708 


981625 


63 


472995 


771 


627005 


27 


34 


455044 


707 


981587 


63 


473457 


770 


520543 


26 


35 


455469 


707 


981549 


63 


473919 


769 


626081 


25 


36 


455893 


706 


981612 


63 


474381 


769 


625619 


24 


37 


456316 


705 


981474 


63 


474842 


768 


525158 


23 


38 


456739 


704 


981436 


63 


475303 


767 


524697 


22 


39 


457162 


704 


981399 


63 


475763 


767 


524237 


21 


40 
41 


457584 


703 


981361 


63 
63 


476223 
9.476683 


766 

765 


623777 


20 
19 


9.458006 


702 


9.981323 


10.523317 


42 


458427 


701 


981285 


63 


477142 


765 


622858 


18 


12 


458848 


701 


981247 


63 


477601 


764 


522399 


17 


459268 


700 


981209 


63 


478059 


763 


521941 


16 


45 


459688 


699 


981171 


63 


478517 


763 


621483 


15 


46 


460103 


698 


981133 


64 


478975 


762 


521025 


14 


47 


460527 


698 


981095 


64 


479432 


761 


620568 


13 


48 


460046 


697 


981057 


64 


479889 


761 


620111 


12 


49 


461364 


696 


981019 


64 


480345 


760 


619655 


11 


50 
51 


461782 


695 


980Q81 
9.980942 


64 
64 


480801 


759 


519199 


10 
9 


9.462199 


695 


9.481257 


759 


10.518743 


52 


462616 


694 


980904 


64 


481712 


758 


618288 


8 


53 


463032 


693 


980866 


64 


482167 


757 


617833 


7 


54 


463448 


693 


980^27 


64 


482621 


757 


617379 


6 


55 


463864 


692 


980789 


64 


483075 


756 


516925 


5 


56 


464279 


691 


980750 


64 


483529 


755 


516471 


4 


57 


464694 


690 


9807 12 


64 


483982 


755 


616018 


3 


58 


465108 


690 


980673 64 


484435 


754 


615565 


2 


59 


465522 


689 


980635; 04 


484887 


7.'^ 3 


515113 


1 


60 


465935 


688 


98059b: 64 


485339 


753 


5vr.ni 





1 


Coaine 1 




1 Sine 


Cotung. J 




1 Tang. M. | 



73 Degrees. 



AKD TJJKiBSTS. (l7 D cg T CO ,) 



3fl 



F 


Siiie 1 


D 1 


Ctmie 1 D. 


1 TMir. 1 


1*. 1 


ComniE. 1 





9.46:>935 


688 


9.98059(i 


64 


9.485;S39 


755 


10.514661160 


1 


466346 


688 


9805.'»8 


64 


485791 


752 


514209 59 


% 


466761 


687 


980519 


65 


486242 


751 


513758 


58 


8 


467173 


. 686 


980480 


65 


486693 


751 


613307 


57 


4 


467585 


685 


980442 


65 


487143 


750 


612857 


56 


6 


467996 


685 


980403 


65 


487593 


749 


512407 


55 


6 


468407 


684 


980364 


65 


488043 


749 


511957 


54 


7 


468817 


683 


980325 


65 


488492 


748 


511508 


53 


8 


469227 


683 


980286 


65 


488941 


747 


511059 


52 


9 


469637 


682 


980247 


65 


489390 


747 


510610 


61 


10 
11 


470046 


681 


980208 


65 
65 


489838 


746 


610162 
10 509714 


60 


9.470455 


680 


9.980169 


9.490286 


746 


49 


12 


470863 


680 


980130 


65 


490733 


745 


509267 


48 


13 


471271 


679 


980091 


65 


491180 


744 


508820 


47 


U 


471679 


678 


980052 


65 


491627 


744 


508373 


46 


15 


472086 


678 


980012 


65 


492073 


743 


607927 


45 


16 


472492 


677 


979973 


65 


492519 


743 


507481 


44 


17 


472998 


676 


979934 


66 


492965 


742 


507035 


43 


18 


473304 


676 


979895 


66 


493410 


741 


506690 


42 


19 


473710 


675 


979865 


66 


493854 


740 


606146 


41 


80 

SI 


474115 


674 


979816 
9.979776 


66 
66 


494299 


740 


505701 


40 


9.474519 


674 


9.494743 


74'. 


10.505267 


39 


22 


474923 


673 


979737 


66 


495186 


739 


604814 


38 


23 


47.5327 


672 


979697 


66 


495630 


738 


604370 37 


24 


475730 


672 


979658 


66 


496073 


737 


503927 


36 


25 


476133 


671 


979618 


66 


49C515 


737 


603485 


35 


26 


476536 


670 


979579 


66 


496957 


736 


603043 


34 


37 


476938 


669 


979539 


66 


497399 


736 


602601 


33 


28 


477340 


669 


979499 


66 


497841 


785 


602169 


32 


29 


477741 


668 


979459 


66 


498282 


734 


601718 


31 


30 
31 


478142 


667 
667 


9794^0 


66 
66 


498722 


734 


601278 


30 


9.478542 


9.979380 


9.499163 


733 


10.500837 


29 


32 


478942 


666 


979340 


66 


499603 


733 


600397 


28 


33 


479342 


665 


979300 


67 


600042 


732 


499958 


27 


34 


479741 


665 


979260 


67 


600481 


781 


499619 


26 


35 


480140 


664 


979220 


67 


600920 


731 


499080 


25 


36 


480539 


653 


979180 


67 


501359 


730 


498641 


24 


37 


480937 


663 


979140 


67 


501797 


730 


498203 


5?3 


38 


481334 


662 


979100 


07 


602235 


729 


497765 


22 


39 


481731 


661 


979059 


67 


502672 


728 


497328 


21 


40 
41 


482128 


66i 


979019 
9.978979 


67 
67 


503109 


728 


496891 
To. 496464 


20 


9.482525 


660 


9.503646 


727 


19 


42 


482921 


659 


978939 


67 


603982 


727 


496018 


18 


43 


483316 


659 


978898 


67 


604418 


726 


496682 


17 


44 


483712 


658 


978858 


67 


504864 


725 


495146 


16 


46 


484107 


657 


978817 


67 


605289 


725 


494711 


15 


46 


484501 


667 


978777 


67 


505724 


724 


494276 


14 


47 


484895 


656 


978736 


67 


606169 


724 


493841 


13 


48 


485289 


656 


978696 


68 


506593 


723 


493407 


12 


49 


485682 


655 


978655 


68 


607027 


722 


492973 


11 


50 
51 


486076 
9.486467 


654 


978615 


68 
68 


607460 


722 


492.540 


10 


653 


9.978674 


9.507893 


721 


10.492107 


9 


52 


486860 


653 


978533 


68 


6C»i326 


721 


491674 


8 


53 


487251 


652 


978493 


68 


608769 


720 


491241 


7 


54 


487643 


651 


978452 


68 


609191 


719 


490809 


6 


55 


488034 


661 


978411 


68 


609622 


719 


490378 


5 


56 


488424 


650 


978370 


68 


610054 


718 


489046 


4 


57 


488814 


650 


9783*^9 


68 


510486 


'!8 


489515 


8 


58 


489204 


649 


978288 


68 


610916 


717 


489084 


2 


59 


489.593 


648 


978247 


68 


611346 


710 


488664 1 


60^ 


489982 


648 


978206 


68 


611776 


716 


488224 1 


u 


Coeine 


1 


Sine j 1 


Cotaiig. 




1 'iaiig 


r-i.-» 






18 




711 


(egreei. 







36 


(18 Degrees.) a 


TABUi OF LOGABmmiC 






1 Pjne 


t n. 


1 Co«ine | I). 


1 Tang. 


1 D. 


1 Ciitans. 1 





9.489982 


648 


9.978206 


68 


9.611776 


716 


10.488224 


60 


1 


490371 


648 


978165 


68 


512206 


716 


487794 


59 


2 


490759 


647 


978124 


68 


512635 


715 


487365 


58 


3 


491147 


646 


978083 


69 


613064 


714 


486936 


67 


4 


491535 


646 


978042 


69 


513493 


714 


486507 


66 


6 


491922 


645 


978001 


69 


513921 


713 


486079 


65 


6 


492308 


644 


977959 


69 


514349 


713 


485651 


64 


7 


492695 


644 


977918 


69 


614777 


712 


485223 


53 


8 


493081 


643 


977877 


69 


615204 


712 


484796 


62 


9 


493466 


642 


977835 


69 


615631 


711 


484369 


61 


10 
11 


493851 


642 


977794 
9.977752 


69 
69 


616057 


710 


483943 


50 
49 


9.494236 


,641 


9.516484 


710 


10.48.3516 


12 


494621 


641 


977711 


69 


616910 


709 


483090 


48 


13 


495005 


640 


977669 


69 


617336 


709 


482665 


47 


14 


495388 


639 


977628 


69 


617761 


708 


482239 


46 


15 


495772 


639 


977.586 


69 


618185 


708 


481815 


46 


16 


496154 


638 


977544 


70 


618610 


707 


481390 


44 


17 


496537 


637 


977503 


70 


619034 


706 


480966 


43 


18 


496919 


637 


977461 


70 


619458 


706 


480542 


42 


19 


497301 


636 


977419 


70 


619882 


705 


480118 


41 


20 
21 


497682 


636 


977377 


70 
70 


520305 


705 


479695 


40 
39 


9.498064 


635 


9.977335 


9.520728 


704 


10.479272 


22 


498444 


634 


977293 


70 


521151 


703 


478849 


38 


23 


498825 


634 


977251 


70 


521573 


703 


478427 


37 


24 


499204 


6.33 


977209 


70 


621995 


703 


478005 


36 


25 


499584 


632 


977167 


70 


622417 


702 


477583 


36 


26 


499963 


632 


977125 


70 


622838 


702 


477162 


34 


27 


500342 


631 


977083 


70 


623259 


701 


476741 


33 


28 


600721 


631 


977041 


70 


623680 


701 


476820 


32 


29 


501099 


6.30 


976999 


70 


624100 


700 


475900 


31 


30 
31 


601476 


629 


976957 


70 
70 


624520 


699 


475480 


30 
29 


9.501854 


629 


9.976914 


9.524939 


699 


10.475061 


32 


502231 


628 


976872 


71 


525359 


'698 


474641 


28 


33 


502607 


628 


976830 


71 


525778 


698 


474222 


27 


34 


502984 


627 


976787 


71 


626197 


697 


473803 


26 


35 


503360 


026 


976745 


71 


626615 


697 


473385 


26 


36 


503735 


626 


976702 


71 


627033 


696 


472967 


24 


37 


504110 


625 


976660 


71 


627451 


696 


472549 


23 


38 


504485 


625 


976617 


71 


527868 


695 


472132 


22 


39 


504860 


624 


976574 


71 


528285 


695 


471715 


21 


40 
41 


505234 

9.505608 


623 


976532 
9.976489 


71 
71 


528702 


694 


471VM8 


20 
19 


623 


9.529119 


693 


0.470881 


12 


505981 


622 


976446 


71 


529535 


693 


470465 


18 


43 


506354 


622 


976404 


71 


529950 


693 


470050 


17 


44 


506727 


621 


9763G1 


71 


530366 


692 


469634 


16 


45 


507099 


620 


976318 


71 


530781 


691 


469219 


15 


46 


507471 


620 


976275 


71 


531196 


691 


468804 


14 


47 


507843 


619 


976232 


72 


531611 


690 


468389 


13 


48 


508214 


619 


976189 


72 


532025 


690 


467975 


12 


49 


508585 


618 


976146 


72 


532439 


689 


467561 


U 


50 
51 


508956 


618 


976103 


72 
72 


532853 


689 


467147 


10 
9 


9.509326 


617 


9.976060 


9.533266 


688 


10.466734 


52 


609696 


616 


976017 


72 


533679 


688 


466321 


8 


53 


510065 


616 


975974 


72 


534092 


687 


465908 


7 


54 


510434 


615 


975930 


72 


534504 


687 


465496 


6 


55 


610803 


615 


975887 


72 


534916 


686 


465084 


6 


56 


511172 


614 


975844 


72 


535328 


686 


464672 


4 


57 


511540 


613 


975800 


72 


535739 


685 


464261 


3 


C-S 


511907 


613 


975757 


72 


636150 


685 


463850 


2 


69 


612275 


612 


975714 


72 


636561 


684 


463439 


1 


60 


512642 


C12 


975670 


72 


536972 


6«4 


463028 





_j 


Cosine 




Sine I 


(;oiaii{!. 




Tang. 1 51. | 



71 Decrees. 





K 




87 




JL 


Sinn 


■d. 


C,»lne 1 D. 


T,,^, 1 D. 1 !.„„„. ( 






9.5l3(i4i 


613 


9.975670 




9.53697^ 


em 


10.463038 








5i3«09 




975627 


73 


637383 


683 


463618 








51337S 


Bll 


975583 


73 


537792 


683 


463308 


SB 






61374! 


610 


975539 


73 


638202 




461798 


57 






614107 


609 


9754BS 


73 


638611 




461389 


5S 






514472 


609 


975452 


73 


639030 


681 


460980 


55 






614837 


608 


9/5408 


73 


639429 




460571 


64 






615302 


608 


975365 


73 


639837 




460163 


63 






515S66 


607 


975331 


T3 


640245 


680 


459766 


63 






615930 


607 


975377 


73 


540B53 


679 


459347 


6! 






616391 


606 


975333 


73 


5410B1 


679 


458939 


50 




9.616857 


605 


9.975189 


9.5414B8 


678 








517020 


605 


975145 




54 1875 


678 




48 






517383 


604 


975101 




643381 


677 


457719 


47 






617746 


eU4 


976067 




513688 


677 










618107 




975013 




543094 


676 


456906 








618468 


603 


974969 


H 


543499 


676 


456501 








518839 


eus 


974925 




543906 


675 


456095 


13 






619190 


BOl 


974880 


74 


544310 


676 


456690 


13 






519551 


GUI 


974836 


74 


544715 


674 


456385 


11 






5199(1 


&tio 


974792 


74 

74 


546119 


674 

673 


4.^4881 
10.451476 


40 
39 




9.63D271 


600 


9.974748' 






2S 


620631 


599 


974703 


74 




673 


154072 


38 




23 


62099'! 


599 


974659 


74 




673 


15366B 


37 




S4 


621349 


59S 


974614 


74 


546735 


673 


15336.i 


38 




as 


631797 


594 


974570 


74 




671 


453862 


35 




se 


622066 


597 


974635 


74 




671 


453160 


34 




S7 


622424 


593 


974481 


74 


517943 


670 


453057 


33 








696 


974436 


71 


548345 


670 


4616.56 


33 




SB 






974391 


74 


548747 




451353 


31 




30 
31 




595 




75 


649149 


669 


450851 


30 

39 




9.6 3853 


S94 


9.974302 


9.5 9550 


668 


10.4504.'b0 




3S 


5 4308 


594 




76 


5 951 


668 


460049 


38 




33 


5 45B4 


5!)3 


974212 


75 


650363 


667 


419648 


37 




34 


6 4930 


693 


974167 


76 


660763 


667 


419248 


36 




3S 


52587S 


592 


974133 


75 


6 163 






as 




3S 


535630 


591 


974077 


76 


6 S63 


666 


448448 


24 




37 


S 5984 


591 


974032 


75 


6 952 


665 


448948 


23 




38 


6 633a 


590 


973987 


76 


5 2361 


665 


447649 






39 




590 


973942 


75 


5 2760 


665 


447250 






40 


6 7046 


589 


973H07 


75 


6 3149 


664 


446851 


20 




*T 


9.527400 


589 


9.973S53 


76 


0.5 3548 


064 


10.116153 


19 




43 


837753 


688 


973507 


75 


5 3946 


663 


416054 


18 




43 


538105 


688 


973761 


76 


5'i4344 


663 


44566B 






44 


538453 


587 


973716 


78 


5j4741 


662 


446359 


18 




45 


538810 


587 


973671 


76 


5 130 


663 


444861 


15 




48 


539161 


586 


973625 


76 




661 


444461 


U 




47 


6S3613 


586 


973,580 


76 


6 93 


661 


444067 


13 




48 


639864 


585 


973535 


76 


6 32 


660 


413671 






49 


530215 


685 


973489 


76 












SO 


I.3D565 


584 


973444 


76 


557l2i 






10 




6T 


I. 5I0-9I5 


584 


B.9rd398 


78 


9. 7617 


^059" 


10.442483 


"6 




fi2 


631265 


583 


97335: 


76 


7913 


669 


443087 


8 




53 


531614 


583 




76 


58308 


668 


441692 


7 




64 


631963 


682 


97336- 


76 


8703 


658 


441398 


S 




S5 


632312 


681 


973315 


76 


9097 


857 


440903 


6 




66 


633661 


681 


973199 


76 


9491 


857 


440500 


4 




67 


633009 


580 


973134 


78 


9885 


656 


440115 


3 




6S 


533357 


6S0 


973078 


76 


0379 


656 


439731 


3 




69 


633704 




973032 


77 


0673 


655 


439337 


1 




eo 


634063 




972986 


77 


1066 


655 


438934 







1 f:.*uw 


^^^ 


di^>o ( 


(.Mcnni, 




T-^ |M- 





98 


^20 Degrees.) a 


TABLE OF LOOABITHHIO 




M. 


1 Bine 


1 D. 


1 Cosine | U. 


Tang. 


1 D. 


1 Cotane. | ] 





9.534052 


678 


9.972986 


77 


9.561066 


655 


10.438y34 


60 


1 


534399 


577 


972940 


77 


661459 


654 


4.38541 


59 


2 


634745 


577 


972894 


77 


561851 


654 


438149 


68 


3 


535092 


577 


972848 


77 


562244 


653 


437756 


57 


4 


635438 


676 


972802 


77 


562636 


653 


437364 


56 


5 


635783 


576 


972755 


'77 


563028 


653 


436972 


65 


6 


536129 


676 


972709 


77 


563419 


652 


436581 


54 


7 


536474 


674 


972663 


77 


563811 


652 


436189 


53 


8 


536818 


674 


972617 


77 


564202 


651 


435798 


52 


9 


637163 


673 


972570 


77 


564592 


651 


435408 


61 


10 
11 


537507 


673 


972524 


77 
77 


564983 


650 


435017 


50 
49 


9.537851 


572 


9.972478 


9.66537.^ 


650 


10.434627 


12 


533194 


672 


972431 


78 


565763 


649 


434237 


48 


13 


638538 


671 


972385 


78 


666153 


649 


433847 


47 


14 


533880 


571 


972338 


78 


666542 


649 


433458 


46 


15 


539223 


570 


972291 


78 


566932 


648 


433068 


45 


16 


539565 


570 


972245 


78 


667320 


648 


432680 


44 


17 


539907 


669 


972198 


78 


567709 


647 


432291 


43 


18 


540249 


569 


972151 


78 


568098 


647 


431902 


42 


19 


640590 


568 


972105 


79 


568486 


646 


431514 


41 


20 
21 


540931 


568 


972058 


78 
78 


568873 


646 


431127 


40 
39 


9.541272 


567 


9.972011 


9,569261 


645 


10.430739 


22 


541613 


567 


971964 


78 


569648 


645 


430352 


38 


23 


641953 


566 


971917 


78 


570035 


645 


429965 


37 


24 


542293 


566 


971870 


78 


670422 


644 


429578 


36 


25 


642632 


565 


971823 


78 


570809 


644 


429191 


35 


26 


542971 


565 


971776 


78 


571195 


643 


428805 


34 


27 


543310 


564 


971729 


79 


571581 


643 


428419 


33 


28 


543649 


564 


971682 


79 


571967 


642 


428033 


^ 


29 


543987 


563 


971635 


79 


572352 


642 


427648 


31 


30 
31 


541325 
9.544663 


563 


971588 


79 
79 


572738 


642 


427262 


30 
29 


562 


9.971540 


9.573123 


641 


10.426877 


32 


545000 


562 


971493 


79 


573507 


641 


426493 


28 


33 


545338 


561 


071446 


79 


573892 


640 


426108 


27 


34 


645674 


561 


971398 


79 


574276 


640 


425724 


26 


35 


546011 


560 


971351 


79 


574660 


639 


425340 


26 


36 


546347 


560 


971303 


79 


575044 


639 


424956 


24 


37 


546683 


559 


971256 


79 


575427 


639 


424573 


23 


38 


547019 


559 


971208 


79 


575810 


638 


424190 


22 


3^ 


547354 


558 


971161 


79 


576193 


638 


423807 


21 


40 
41 


547689 


558 


971113 
9.971066 


79 

80 


576576 


637 


423424 


20 
19 


9.548024 


557 


9.576958 


637 


10.423041 


42 


548359 


557 


971018 


80 


577341 


636 


422659 


18 


43 


548693 


556 


970970 


80 


577723 


636 


422277 


17 


44 


549027 


550 


970922 


80 


578104 


636 


421896 


16 


45 


549360 


555 


970874 


80 


578486 


635 


421514 


15 


46 


549693 


555 


970827 


80 


578867 


635 


421133 


14 


47 


650026 


554 


970779 


80 


579248 


634 


420752 


13 


48 


550359 


554 


970731 


80 


579629 


634 


420371 


12 


49 


650692 


553 


970683 


80 


530009 


634 


419991 


11 


50 
51 


551024 
9.551356 


553 


970635 
9.970586 


80 
80 


580389 


633 


419611 


10 
9 


552 


9.580769 


633 


10.419231 


52 


551087 


552 


•970538 


80 


581149 


632 


418851 


8 


53 


552018 


552 


970490 


80 


581528 


632 


418472 


7 


54 


552349 


551 


970142 


80 


581907 


632 


418093 


6 


55 


552680 


551 


970394 


80 


582286 


631 


417714 


5 


56 


553010 


550 


970345 


81 


582665 


631 


417335 


4 


57 


553341 


550 


970297 


81 


583043 


630 


416957 


3 


58 


553670 


549 


970249 


81 


683422 


630 


416578 


2 


59 


554000 


549 


970200 


81 


583800 


629 


416200 


1 


60 


554329 


54S 


970152 


81 


584 177 


029 


415S23 





L 


1 Co.siuii 




riint 1 


COlvTUp. j 




j Tansr. M. | 



09 Degrees. 



BIKES Aia> TANOEXrs. (21 Degtccs.j 



39 



1. 1 Siiie I D. I C^»^i■ *- I D. I 'r;i..g. | D. | Cotaiie | 







1 
2 
3 

4 
5 
6 
7 
8 
9 
10 

11 
12 
13 
14 
15 
16 
17 
18 
19 

^ 
21 
22 
23 
24 
25 
26 
27 
28 
29 
30 

31 
32 
83 
34 
35 
36 
37 
38 
39 
40 

41 
42 
43 
44 
45 
46 
47 
4'< 
49 
50 

51 
52 
53 
54 
55 
56 
57 
58 
59 
60 



9.5o432^ 
554658 
554987 
555315 
555643 
555971 
556299 
556626 
556953 
557280 
55760C 



9.557932 
559258 
558583 
558909 
559234 
559558 
559883 
660207 
560531 
560855 



.561178 
561501 
561824 
5C2146 
562468 
562790 
563112 
563433 
563755 
564075 



548 I 9. 

548 

547 

547 

546 

546 

545 

545 

544 

544 ] 

543 



543 
543 
542 
542 
541 
541 
540 
540 
539 
539 



9. 



,564396 
564716 
565036 
565350 
565076 
565995 
566314 
566632 
566951 
567269 



9.567587 
567904 
568222 
568539 
568856 
569172 
509488 
569804 
570120 
570435 



9.570751 
571066 
571380 
571695 
572009 
572323 
572636 
572950 
573263 
573575 



538 
538 
537 
537 
536 
536 
536 
535 
535 
534 



534 
533 
533 
532 
532 
531 
531 
531 
530 
W0_ 

529 

529 
528 
528 
528 
527 
527 
526 
526 
_525 

525' 

524 
524 
523 
523 
523 
522 
522 
521 
.521 



9'i0io2 
970103 
970055 
970006 
969957 
969909 
969860 
969811 
969762 
969714 
969665 



9 



969616 
969567 
969518 
969469 
969420 
969370 
969321 
969272 
969223 
909173 



9. 



969124 
969075 
969025 
968976 
968926 
968877 
968827 
968777 
968728 
968678 



.968628 
968578 
968528 
968479 
968420 
968379 
968^29 
968:i78 
968228 
968178 

.968128 
968078 
968027 
967977 
967927 
967876 
967820 
907775 
967725 

_967674 

.967024 
907573 
907.522 
967471 
G07421 
9C7370 
967319 
967268 
967217 
967166 



»1 
81 
81 
81 
81 
81 
81 
81 
81 
81 

§1 
82 
82 
82 
82 
82 
82 
82 
82 
82 
82 

82 
82 
82 
82 
83 
83 
83 
83 
83 
83 

83 
83 

83 
83 
83 
83 

83 
83 

84 
84 

84 
84 
84 
84 
84 
84 
84 
84 
84 
84 

84 
84 
85 
85 
85 
85 
85 
85 
85 
85 



9.5«4177 
584555 
584932 
685309 

. 5850{S6 
586062 
586439 
586815 
587190 
587566 
587941 

9.588316 
588691 
589066 
589440 
589814 
590188 
590562 
590935 
.'^91308 
59168 1 

9.59*>^054 
592426 
592798 
593170 
593542 
593914 
594285 
594656 
596027 
595398 



9.595768 
596138 
596508 
596878 
597247 
597616 
597985 
598354 
598722 
599091 

9.599459 
599827 
600194 
600562 
600929 
60r^9« 
60160-2 
602029 
60239O 
60276 1 

9.603127 
60.3493 
603858 
604223 
604588 
604953 
605317 
605682 
606046 
606410 



629 
629 
628 
628 
627 
627 
627 
626 
626 
625 
625 



625 
024 
624 
623 
623 
623 
622 
622 
622 
621 



621 
620 
620 
619 
619 
618 
618 
618 
617 
617 



617 
616 
616 
616 
615 
615 
615 
614 
614 
613 



613 
613 
613 
612 
611 
611 
611 
610 
610 
JIO 

609' 

609 

609 

608 

608 

607 

607 

607 

606 

606 



10.415823 
415445 
4].'>068 
414691 
414314 
413938 
413561 
4131^5 
412810 
412434 

412059 

10.411684 
411309 
410934 
410560 
410186 
409812 
409438 
409065 
408692 
4083 19 

10.40794(5 
407574 
407202 
406829 
406458 
406086 
405715 
405344 
404973 

404602 

10.404^32 
403802 
403492 
403122 
402763 
402384 
402015 
401646 
401278 
400909 



10.400541 
400173 
899806 
399438 
399071 
398704 
398338 
397971 
397605 

39723 9 

10.396873 
396.507 
390142 
895777 
395412 
395047 
394083 
394318 
39395^ 
3935r 



60 
59 
58 
57 
56 
55 
51 
53 
52 
51 
50 

49 
48 
47 
46 
45 
44 
43 
42 
41 
40 

39 
38 
37 
36 
35 
34 
33 
32 
31 
30 

29 
28 
27 
26 
25 
24 
23 
22 
21 
20 

19 
18 
17 
16 
15 
14 
13 
12 
11 
10 

9 

8 
7 
6 
5 
4 
3 



Zl 



Co-ne 



I 



I »'^».«e I I C.itaiijr. I 



18* 



{ 'J'aiif 






40 


(22 Degr 


ees.} A 


TABJ«JS OF LOoARirnmc 




M. 


1 sine 


1 D. 


1 Ciwine 1 D. 


1 Tansr. 


1 D. 


Collins. 







9. 573575 


521 


9.967166 


85 


9.606410 


606 


10.393590 


60 


1 


673888 


520 


967115 


85 


606773 


606 


393227 


59 


2 


674200 


520 


967064 


85 


607137 


605 


392863 


58 


3 


674512 


519 


967013 


85 


607500 


605 


892500 


67 


4 


674824 


519 


966961 


85 


607863 


604 


392137 


56 


5 


576136 


519 


966910 


85 


G08225 


604 


391775 


55 


6 


575447 


518 


966859 


85 


608588 


604 


391412 


54 


7 


575758 


518 


966808 


85 


608950 


603 


891050 


53 


8 


676069 


517 


966756 


86 


609312 


603 


390688 


52 


9 


576379 


517 


966705 


86 


1B09674 


603 


890.326 


51 


10 
11 


576689 


516 


966653 
9.966G02 


86 
86 


610036 
9.610397 


602 


389964 
10.389603 


.50 
49 


9.576999 


516 


602 


12 


677309 


516 


966550 


86 


610759 


602 


389241 


48 


13 


577618 


515 


966499 


86 


611120 


601 


388880 


47 


14 


577927 


515 


966447 


86 


611480 


601 


388520 


46 


15 


578236 


514 


966395 


86 


611841 


601 


888159 


45 


16 


578545 


514 


966344 


86 


612201 


600 


.387799 


44 


17 


578853 


513 


966292 


86 


612561 


600 


887439 


43 


18 


579162 


513 


966240 


86 


612921 


600 


887079 


42 


19 


579470 


513 


966188 


86 


613281 


699 


386719 


41 


20 
21 


579777 


512 


966136 


86 
87 


613641 


.599 


886359 


40 
39 


9.580085 


512 


9 966085 


9.614000 


598 


10.386000 


22 


580392 


511 


966033 


87 


614359 


598 


38.5641 


88 


23 


580699 


511 


966981 


87 


6^4718 


598 


38.5282 


37 


24 


581005 


511 


965928 


87 


616077 


597 


384923 


36 


25 


.581312 


510 


965876 


87 


615435 


597 


384565 


35 


26 


681618 


510 


965824 


87 


61.5793 


597 


384207 


34 


27 


581924 


509 


965772 


87 


616151 


596 


883849 


38 


28 


582229 


.509 


965720 


87 


616.509 


596 


383491 


82 


29 


582535 


509 


965668 


87 


616867 


596 


883133 


81 


30 
31 


.582840 


508 


965615 


87 

87 


617224 


595 


382776 
10.. 382418 


30 
29 


9.583145 


508 


9.965563 


9 617582 


595 


32 


583449 


.507 


965511 


87 


617939 


595 


382061 


28 


33 


583754 


507 


965458 


87 


618295 


594 


881705 


27 


34 


584058 


506 


965406 


87 


618652 


594 


881348 


5>6 


35 


584361 


506 


965353 


88 


619008 


594 


380992 


25 


36 


584665 


•^oe 


965301 


88 


619364 


593 


380636 


24 


37 


584968 


<505 


965248 


88 


619721 


593 


380279 


23 


38 


585272 


505 


965195 


88 


620076 


593 


379924 


22 


39 


585574 


504 


965143 


88 


620432 


592 


879568 


21 


40 

41 


585877 


504 


965090 
9.905037 


88 
88 


620787 
9.621142 


592 


379213 


20 
19 


9.. 586 179 


503 


592 


10.376858 


42 


586482 


503 


964984 


88 


621497 


591 


378503 


18 


43 


586783 


503 


964931 


88 


621852 


591 


378148 


17 


44 


587085 


502 


964879 


88 


622207 


590 


877793 


16 


45 


587386 


502 


964826 


88 


622561 


590 


877439 


15 


46 


587688 


501 


964773 


88 


622915 


590 


377085 


14 


47 


.587989 


501 


^64719 


88 


623269 


.589 


376731 


13 


48 


588289 


.501 


964666 


89 


623623 


589 


376377 


12 


49 


588590 


500 


964613 


89 


623976 


589 


376024 


11 


50 
51 


588890 
9.. 589 190 


500 


964560 
9.964507 


89 
89 


624330 
9.624683 


588 


375670 


10 
9 


499 


588 


10.375317 


52 


589489 


499 


964454 


89 


625036 


588 


374964 


8 


53 


.589789 


499 


964400 


89 


625388 


.587 


874612 


7 


54 


5900S8 


498 


964347 


89 


625741 


587 


374259 


6 


55 


5903S7 


498 


964294 


89 


626093 


587 


373907 


5 


56 


590686 


497 


964240 


89 


626445 


586 


873555 


4 


57 


590984 


497 


964187 


89 


626797 


586 


373203 


3 


58 


591282 


497 


9641.33 


89 


627149 


.'586 


372851 


2 


59 


591580 


406 


964080 


89 


627,501 


585 


372499 


1 


60 


.591878 


496 


964026^89 


627852 


585 


372148 







Cosine 1 




8i .»; 


C«t;iiia. j 




1 'J^aiig. 1 M. 1 



67 PeKrcM"*. 



SINES A^D TANGKNi's. (?3 Degrecs.j 



41 



M. 


1 Sine 


1 n. 


1 C-sino 1 D. 


Tarn 


D. 


1 Cotaujr. 


"" 





9.691878 


496 


9.96405i6 


89 


9.627852 


585 


10T372T48" 


-60' 


1 


692176 


495 


963972 


89 


628203 


685 


371/97 


69 


2 


692473 


496 


963919 


89 


628664 


585 


371446 


68 


3 


692770 


495 


963865 


90 


628905 


684 


371096 


67 


4 


693067 


494 


963811 


90 


629256 


684 


370745 


66 


5 


693363 


494 


963757 


90 


629606 


683 


370394 


65 


6 


593659 


493 


963704 


90 


629956 


583 


370044 


64 


7 


6939:i6 


493 


963650 i 90 


630306 


583 


369694 


63 


8 


694251 


493 


963596190 


630666 


683 


369344 


62 


9 


694M7 


492 


963543! 90 


631006 


682 


368995 


61 


10 
11 


694842 


492 


963488 


90 
90 


631365 


682 


368646 


50 


9.595137 


491 


9.963434 


9.631704 


682 


10.368296 


49 


12 


595432 


491 


963379 


90 


632053 


581 


367947 


48 


13 


696727 


491 


96.'5325 


90 


632401 


581 


367599 


47 


14 


696021 


490 


963271 


90 


633760 


581 


367250 


46 


15 


690316 


490 


963217 


90 


633098 


580 


.366902 


45 


16 


696609 


489 


963163 


90 


633447 


680 


366553 


44 


17 


696903 


489' 


963108 


91 


633796 


680 


366205 


43 


18 


697196 


489 


963054 


91 


634143 


679 


365857 


42 


19 


697490 


488 , 


962999 


91 


634490 


679 


365510 


41 


20 
21 


597783 


488 


962945 


91 
91 


6.14838 
9.035185 


579 


365162 


40 


9.698075 


487 


9.962890 


578 


10.364815 


39 


22 


698368 


487 


962836 


91 


635532 


678 


364468 


38 


23 


698660 


487 


962781 


91 


635879 


678 


364121 


37 


24 


698952 


486 


962727 


91 


63G226 


677 


363774 


36 


*«5 


599244 


486 


962672 


91 


636572 


577 


363428 


35 


26 


699536 


485 


962617 91 


636919 


577 


30.3081 


.34 


27 


699827 


485 


962562 


91 


637265 


677 


362736 


33 


28 


600118 


485 


962508 


91 


637611 


676 


362389 


32 


29 


600409 


484 


962453 


91 


637956 


676 


,362044 


31 


30 
31 


600700 


484 


962398 


92 
92 


63S302 


676 


361698 


30 


9.600990 


484 


9.962343 


9.638647 


676 


10.361353 


29 


32 


601280 


483 


962288 


92 


638992 


675 


361008 


28 


33 


601570 


483 


962233 


92 


639337 


675 


360663 


27 


34 


601860 


482 


962178 


92 


639682 


574 


360318 


26 


35 


602150 


482 


962123 


92 


640027 


674 


369973 


26 


36 


602439 


482 


962067 


92 


640371 


574 


359629 


24 


37 


602728 


481 


962012 


92 


640716 


673 


359284 


23 


38 


603017 


481 


961957 


92 


641060 


573 


358940 


22 


39 


603305 


481 


961902 


92 


641404 


573 


358596 


,21 


40 
41 


603594 


480 


961846 


92 
92 


641747 


572 


368253 


20 


9.603882 


480 


9.961791 


9.642091 


572 
572 


10.357909 


19 


42 


604170 


479 


961735 


92 


642434 


367666 


18 


43 


604457 


479 


961680 


92 


642777 


572 


367223 


17 


44 


604746 


479 


961624 


93 


643120 


571 


366880 


16 


45 


605032 


478 


961669 


93 


64.S'i63 


571 


366637 


15 


46 


605319 


478 


961513 


93 


643806 


571 


356194 


14 


47 


606606 


478 


961458 


93 


644148 


570 


365862 


13 


48 


605892 


477 


961402 


93 


644490 


570 


35.5610 


12 


49 


606179 


477 


961346, 93 


644832 


570 


366168 


11 


50 

61 


606466 


476 


961290 


93 
93 


645174 


669 


364826 


10 


9 606751 


476 


9.961236 


9.645616 


569 


10.364484 


9 


52 


607036 


476 


961179 


93 


646867 


569 


364143 


8 


53 


607322 


475 


961123 


93 


646199 


569 


353801 


7 


54 


607607 


475 


961067 


93 


646540 


568 


363460 





55 


607892 


474 


961011 


93 


646881 


668 


363119 


5 


56 


608177 


474 


960965 


93 


647222 


668 


362778 


4 


57 


608461 


474 


960899 


93 


647562 


567 


362438 


3 


5S 


608746 


473 


960843 


94 


647903 


567 


352097 


2 


59 


609029 


473 


960786 


94 


648243 


567 


351767 


1 


60 


609318 


473 


960730 


94 


648583 


666 


351417 





1 


CiNsiiie 




8tnu 1 


Cotnti(f. 


1 


1 Tn»r. 





MDegreee. 



42 


(24 DegreesO a 


TABLE OF LOOAfilTHHIC 




"iT 


Sine 


1 D. 


1 a»hfne D. 


1 'I'anij. 


1 D. 


Cotftng. 1 1 





9.609313 


473 


9.9607.30 


94 


9.648683 


566 


10. .351417 


60 


1 


609597 


472 


960674 


94 


648923 


566 


351077 


59 


2 


609880 


472 


960618 


94 


649263 


566 


350737 


58 


3 


610164 


472 


960561 


94 


649602 


566 


350398 


57 


4 


610447 


471 


D60606 


94 


649942 


565 


350058 


56 


5 


610729 


471 


960448 


94 


650281 


565 


849719 


55 


6 


611012 


470 


960392 


94 


650020 


565 


349.380 


54 


7 


611294 


470 


960336 


94 


650959 


564 


349041 


53 


8 


611676 


470 


960279 


94 


661297 


564 


348703 


52 





61 1858 


469 


900222 


94 


651636 


564 


348364 


51 


10 
11 


612140 


469 


960165 


94 
95 


651974 


563 


348026 
10.347688 


50 
49 


9.612421 


469 


9.960109 


9.662312 


563 


12 


612702 


468 


960052 


95 


652650 


563 


347350 


48 


13 


612983 


468 


969996 


95 


652988 


663 


347012 


47 


14 


613264 


467 


969938 


95 


6.63326 


562 


346674 


46 


15 


613545 


467 


959882 


95 


653663 


562 


346337 


45 


16 


613825 


467 


959825 


96 


654000 


562 


346000 


44 


17 


614105 


466 


959768 


95 


654337 


561 


345663 


43 


18 


614385 


466 


959711 


95 


654674 


561 


345326 


42 


19 


614665 


466 


959654 


95 


65501 1 


561 


344989 


41 


20 
21 


614944 


465 


959596 


95 
95 


655348 


661 


344652 


40 
39 


9.015223 


465 


9.959639 


9.6.65684 


560 


10.344316 


22 


615502 


465 


959482 


95 


666020 


560 


343980 


38 


23 


615781 


464 


959425 


96 


656356 


560 


343644 


37 


24 


616060 


464 


969368 


95 


656692 


569 


343308 


36 


25 


616338 


464 


959310 


96 


667028 


559 


342972 


35 


26 


616616 


463 


959263 


96 


667364 


559 


342636 


34 


27 


616894 


463 


969196 


96 


657699 


559 


342301 


33 


28 


617172 


462 


969138 


96 


658034 


558 


341966 


33 


29 


617450 


462 


969081 


96 


658369 


558 


341631 


31 


30 
31 


617727 


462 


969023 


96 
96 


658704 


558 


341296 


30 
29 


9.618004 


461 


9.9689C5 


9.659039 


568 


10.340961 


32 


618281 


461 


968908 


96 


669373 


657 


340627 


28 


33 


618558 


401 


968850 


96 


659708 


657 


840292 


27 


34 


618834 


460 


968792 


96 


660042 


567 


339958 


26 


35 


619110 


460 


968734 


96 


660376 


557 


339624 


26 


36 


619386 


460 


968677 


96 


660710 


556 


339290 


24 


37 


619662 


469 


968619 


96 


661043 


556 


338967 


23 


38 


619938 


459 


968661 


96 


661377 


556 


338623 


22 


39 


620213 


459 


968603 


97 


661710 


565 


3.S8290 


21 


40 
41 


620488 


468 


968445 


97 
97 


662043 


655 


337957 
10.337624 


20 

19 


9.620763 


458 


9.968387 


9 662376 


565 


42 


621038 


467 


958329 


97 


662709 


554 


337291 


18 


43 


621313 


467 


* 9,68271 


97 


663042 


554 


336968 


17 


U 


621587 


467 


958213 


97 


663376 


554 


836625 161 


45 


621861 


456 


958164 


97 


663707 


554 


336293 


15 


46 


622135 


456 


968096 


97 


664039 


553 


33.6961 


14 


47 


622409 


466 


968038 


97 


664371 


553 


335629 


13 


48 


622682 


466 


967979 


97 


664703 


563 


335297 


12 


49 


622956 


465 


957921 


97 


665035 


.663 


834965 


11 


50 
51 


623229 


455 


957863 


97 
9V 


666366 
9.666697 


552 


3346.34 


10 
9 


9.623502 


464 


9.9.67804 


562 


10.334303 


52 


623774 


454 


957746 


98 


666029 


562 


333971 


8 


53 


624047 


464 


967687 


98 


666360 


551 


333640 


7 


54 


624319 


463 


957628 


98 


666691 


551 


833309 


6 


55 


624591 


463 


967670 


98 


667021 


561 


832979 


5 


56 


624863 


463 


957511 


98 


667362 


561 


832648 


4 


57 


625135 


462 


967462 


98 


667682 


550 


832318 


3 


58 


626406 


462 


967393 


98 


668013 


560 


331987 


2 


59 


625677 


462 


967336 


98 


668343 


650 


331667 


1 


60 


626918 


461 


967276 98 


668672 


560 


331328 







Cosine 




i Bine | 


Cotani;. 




1 I'ang. 1 M. 1 



65 Degrees. 



SINES AND TANuENTs. (25 Degrecs.l 



43 



M. ( Sine | P. | Cosine | D. | Tang. | D. | 





1 

2 
3 
4 
6 
6 
7 
8 
9 
10 

11 
12 
13 
14 
15 
16 
17 
18 
19 
20 

21 
22 
23 
24 
2i> 
26 
57 
{8 
29 
30 

31 
32 
83 
34 
)5 
8'6 
37 
38 
39 
40 

41 
42 
43 
44 
45 
46 
47 
48 
49 
60 

51 
52 
53 
54 
55 
56 
57 
58 
59 
60 



9.625948 


451 


626219 


451 


626490 


451 


626760 


450 


627030 


450 


627300 


450 


627570 


449 


627840 


449 


628109 


449 


628378 


448 


628647 


448 


9.628916 


447 


629 18r) 


447 


629453 


447 


629721 


446 


629989 


446 


630267 


446 


630524 


446 


630792 


445 


631059 


445 


^31320 


445 


9.631593 


444 


631859 


444 


632126 


444 


632392 


443 


632668 


443 


032923 


443 


633180 


442 


633454 


442 


633719 


442 


633984 


441 


9.034249 


441 


6:M514 


440 


634778 


440 


635042 


440 


63530C 


439 


635570 


439 


635834 


439 


636097 


438 


636360 


438 


636623 


438 
437 


9.038886 


637148 


437 


637411 


437 


637673 


437 


637935 


436 


638197 


436 


638458 


436 


638720 


435 


638981 


435 


639242 


435 


9.639503 


434 


639764 


434 


640024 


434 


640284 


433 


640544 


433 


640804 


433 


641064 


432 


641324 


432 


641584 


432 


641842 


431 



9.9575*76 
957217 
957158 
957099 
95V040 
956981 
956921 
956862 
956803 
956744 
956684 



9.950625 
95656G 
956506 
956447 
956387 
956327 
956268 
956208 
956148 
956089 



9.956029 
955969 
955909 
955849 
955789 
•955729 
955669 
955609 
955548 
955488 



9.955428 
955368 
955307 
955247 
9.55 1 SO 
956126 
955065 
955005 
954944 
954883 

9 954823 
954762 
954701 
954640 
954579 
954518 
954457 
954396 
954335 
954274 



9.954213 
954152 
954090 
9.54029 
953968 
953906 
953845 
953783 
953722 
953660 



9819.668673 

98' 669002 

98 669332 

98 669661 

98 669991 

98 670320 

99 670649 
99 670977 
99 671.306 
99 67 1034 
99 671963 



99 
99 
99 
99 
99 
99 
99 
00 
00 
00 



00 
00 
00 
00 
00 
00 
00 
00 
00 
00 



01 
01 
01 
01 
01 
01 
01 
01 
01 
01 

01 
01 
01 
01 
01 
02 
02 
02 
02 
02 

02 
02 
02 
02 
02 
02 
02 
02 
03 
03 



9.672291 
672619 
672947 
073274 
673602 
673929 
674257 
674584 
674910 
675237 



9. 



675564 
675890 
676216 
676543 
676e'09 
677194 
677520 
677846 
678171 
678496 



9 



678821 
679146 
679471 
679795 
680120 
680444 
680768 
681092 
681416 
681740 



9.682063 
682387 
682710 
683033 
68.3356 
683679 
684001 
684324 
684646 
68496 8 

9.686290 
6856 12 
685934 
686255 
686577 
686898 
687219 
687.'i40 
687861 
688182 



550 
549 
549 
549 
548 
548 
548 
548 
547 
647 
547 



547 
546 
546 
546 
546 
545 
545 
545 
544 
544 



544 
544 
543 
543 
543 
543 
542 
642 
542 
_542 

64T 

641 

541 

541 

540 

540 

640 

640 

639 

539 
539 
538 
638 
638 
538 
537 
537 
537 
537 



536 
536 
536 
536 
C35 
535 
635 
535 
534 
534 



rnrang. ( 1 


10.33I327|6ol 


330998 


m 


330668 


o8 


330339 


57 


3**0009 


66 


329680 


65 


329351 


54 


329023 


53 


328694 


52 


328366 


51 


328037 


50 
49 


10.327709 


327381 


48 


327053 


47 


326726 


46 


326398 


45 


326071 


44 


325743 


43 


325416 


42 


326090 


41 


324763 


40 
39 


10.324436 


3241 10 


38 


323784 


37 


323457 


36 


323131 


35 


322806 


34 


322480 


33 


322154 


32 


321829 


31 


321504 


30 
29 


10.321179 


320854 


28 


320529 


27 


320205 


26 


319880 


25 


319.556 


24 


319232 


23 


318908 


22 


318.584 


21 


318260 


20 
19 


10.317937 


317613 


18 


317290 


17 


316967 


16 


316644 


15 


316321 


14 


315999 


13 


31.5676 


12 


316354 


11 


315032 


10 
9 


10.314710 


314388 


8 


314066 


7 


313745 


6 


313423 


5 


313102 


4 


312781 


3 



312460 

312' 

SV 



I Coaiue I 



I Sine 



I Cutang. I 



I Tw 



04 Decreet. 
FF* 



44 


(2 


6 Degrees.) a 


TABLE OF LOGABITHiaC 




IT 


Sin^ 


D. 


1 Citiiwi 1 D. 


1 TV.. 


D 


1 Ci^tnng, 1 1 





9.641842 


431 


9.963660 


lOb 


9.688182 


534 


r 10.311818 


60 


1 


642101 


431 


953599 


103 


688602 


634 


311498 


69 


2 


642360 


431 


953537 


103 


688823 


634 


311177 


58 


3 


642618 


430 


953475 


103 


689143 


633 


310857 


67 


4 


642877 


i 430 


953413 


103 


689463 


633 


310.537 


66 


5 


643135 


, 430 


953352 


103 


689783 


633 


310217 


65 


6 


643393 


430 


953290 


103 


690103 


633 


309897 


64 


7 


643650 


429 


953228 


103 


690423 


633 


309577 


63 


8 


643908 


429 


95:^66 


103 


690742 


632 


309258 


62 


9 


644165 


429 


953104 


103 


691062 


632 


3089.38 


61 


10 
11 


644423 
9.644680 


428 


953042 


103 
104 


691381 


632 


308619 
10.308300 


50 
49 


428 


9.952980 


9.691700 


631 


12 


644936 


428 


952918 


104 


692019 


631 


307981 


48 


13 


645193 


427 


952855 


104 


692338 


631 


307662 


47 


14 


64.5450 


427 


952793 


104 


692656 


631 


307344 


46 


15 


645700 


427 


9.52731 


104 


692976 


631 


307025 


45 


16 


645962 


426 


952669 


104 


693293 


630 


306707 


44 


17 


646218 


426 


952606 


104 


69.3612 


630 


306388 


43 


18 


646474 


426 


952.544 


104 


693930 


630 


306070 


42 


19 


646729 


425 


9.52481 


104 


694248 


630 


3067.52 


41 


20 


646984 
9.647240 


426 


952419 104 
9.9.52356 104 


694566 
9.694883 


629 


306434 


40 
39 


21 


425 


629 


10.305117 


22 


647494 


424 


952294 


104 


69.5201 


629 


304799 


38 


23 


647749 


424 


952231 


104 


695518 


629 


304482 


37 


24 


648004 


424 


9.52168 


105 


695836 


629 


304164 


38 


25 


648258 


424 


952106 


105 


696153 


628 


303847 


36 


26 


648512 


423 


952043 


105 


696470 


628 


303530 


34 


27 


648766 


423 


951980 


105 


696787 


628 


303213 


39 


28 


649020 


423 


951917 


105i 


697103 


528 


302897 


9St 


29 


649274 


422 


951854 


105 


697420 


627 


302680 


31 


30 
31 


649527 
9.649781 


422 


951791 


106 
105 


697736 


.527 


302264 


30 
29 


422 


9.961728 


5^698053 


527 


10.301947 


32 


650034 


422 


951665 


105 


698369 


527 


301631 


28 


33 


650287 


421 


951602 


105 


698685 


.526 


301316 


27 


34 


650539 


421 


961539 


105 


699001 


626 


300999 


26 


35 


650792 


421 


951476 


105 


699316 


626 


300684 


26 


30 


651044 


420 


951412 


105 


690032 


626 


30036b 


24 


37 


651297 


420 


951349 


106 


699947 


526 


300053 


23 


38 


051.549 


420 


951286 


106 


700263 


525 


299737 


22 


39 


651800 


419 


951222 


106 


700578 


525 


299422 


21 


40 

41 


652052 
9.652304 


419 
419 


9511.59 
9.951096 


106 
106 


700893 


525 


299107 


20 
19 


9.701208 


524 


10.298792 


42 


052555 


418 


951032 


106 


701523 


524 


298477 


18 


43 


652806 


418 


950968 


106 


701837 


524 


298163 


17 


44 


653057 


418 


9.50905 


106 


702152 


524 


297848 


16 


45 


653308 


418 


950841 


100 


702466 


.524 


297534 


15 


46 


653558 


417 


950778 


106 


702780 


.523 


297220 


14 


47 


653308 


417 


9.50714 


106 


703095 


523 


296906 


13 


48 


654059 


417 


9.50650 


106 


703409 


523 


296,591 


12 


49 


654309 


• 416 


950586 


106 


703723 


523 


296277 


11 


50 
51 


654558 
9.654808 


416 


950522 


107 
107 


704036 
9.7013.50 


522 


295964 
10.295650 


10 
9 


416 


9.9.504.58 


522 


52 


65.5058 


416 


950394 


107 


704663 


622 


296337 


8 


53 


65.5307 


415 


950330 


107 


704977 


522 


295023 


7 


54 


655556 


, 415 


950266 


107 


705290 


622 


294710 


6 


55 


655805 


; 415 


950202 


107 


705603 


621 


294397 


6 


56 


656054 


414 


950138 


107 


705916 


621 


294084 


4 


57 


656302 


414 


950074 


107 


70022S 


.521 


293772 


S 


58 


656531 


414 


950010 


107 


706541 


.521 


293459 


2 


59 


656799 


413 


949945 


107 


70685 1 


.521 


293146 


1 


60 


657047 


413 


919881 


107 


707166 


520 


292834 





n 


Cos'iiie 


1 

1 


Sine I 1 


Cdiaim. 




I 'l-.iiig. 1 M. 1 



C3 Degrees. 





SINES AND T4NGENTS. (27 Degrees.) ^ 


45 


M 


Sine 1 


I) 


Cosine j D. 


Tang. 


1 D. 


! Cotnng. I 1 





0.657047 


413 


9.949881 


107 


9.707166 


520 


10.29'-i834 


60 


1 


657295 


413 


949816 


107 


707478 


520 


292622 


59 


2 


667542 


412 


949762 


107 


707790 


520 


292210 


58 


3 


657790 


412 


949688 


108 


708102 


520 


291898 


57 


4 


658037 


412 


949623 


108 


708414 


519 


291586 


56 


5 


658284 


412 


949568 


108 


708726 


519 


291274 


55 


6 


658631 


411 


949494 


108 


709037 


519 


290963 


54 


7 


668778 


411 


949429 


108 


709349 


519 


290661 


53 


S 


669025 


411 


949364 


108 


709660 


519 


290340 


52 


9 


659271 


410 


949300 


108 


709971 


518 


290029 


51 


10 
11 


669617 


410 


949235 


108 
108 


710282 


518 
518 


289718 


50 
49 


9.659763 


410 


9.949170 


9.710693 


10.289407 


12 


660009 


409 


949105 


108 


710904 


518 


289096 


48 


13 


660265 


409 


949040 


108 


711215 


518 


288785 


47 


14 


660601 


409 


948975 


108 


711.625 


517 


288475 


46 


15 


660746 


409 


948910 


108 


711836 


517 


288 164 


45 


16 


660991 


408 


948845 


108 


712146 


517 


287854 


44 


17 


661236 


408 


948780 


109 


712456 


517 


287544 


43 


18 


661481 


408 


948715 


109 


712766 


516 


287234 


42 


19 


661726 


407 


948650 


109 


713076 


516 


286924 


41 


20 
21 


661970 
9.662214 


407 
407 


948584 


109 
109 


7133S6 


516 


286614 


40 
39 


9.948519 


9.713696 


516 


10.286304 


22 


662459 


407 


948464 


109 


714005 


516 


286995 


38 


23 


662703 


406 


948388 


109 


714314 


515 


285686 


37 


24 


662946 


406 


948323 


109 


714624 


^:5 


286376 


36 


25 


663190 


406 


948267 


109 


714933 


5U 


285067 


35 


26 


663433 


405 


948192 


109 


715242 


515 


284768 


34 


27 


663677 


405 


948126 


109 


715551 


514 


284449 


33 


28 


663920 


406 


948060 


109 


715860 


514 


284140 


32 


29 


664163 


405 


947996 


110 


716168 


514 


283832 


31 


30 
31 


664406 


404 


947929 


110 
110 


716477 


514 


283523 


30 
29 


9.664648 


404 


9.947863 


9.716785 


514 


10.283216 


32 


664891 


404 


947797 


110 


717093 


613 


282907 


28 


33 


665133 


403 


947731 


110 


717401 


513 


282699 


27 


34 


665376 


403 


947666 


110 


717709 


513 


282291 


26 


35 


666617 


403 


947600 


110 


718017 


513 


281983 


25 


36 


666869 


402 


947633 


110 


718325 


513 


281670 


24 


37 


660 100 


402 


947467 


uo 


718633 


512 


281367 


23 


38 


606342 


402 


947401 


110 


718940 


512 


281060 


22 


39 


666683 


402 


947336 


110 


719248 


512 


280762 


21 


40 
41 


666834 


401 


947269 


110 

no 


7196.56 


512 


280446 


20 
19 


9.667065 


401 


9.947203 


9.719862 


512 


10.280138 


42 


667306 


401 


947136 


HI 


720169 


511 


279331 


18 


43 


667546 


401 


947070 


111 


720476 


511 


279524 


17 


44 


667786 


400 


947004 


111 


720783 


511 


279217 


16 


45 


668027 


400 


946937 


111 


721089 


511 


278911 


15 


46 


668267 


400 


946871 


111 


721396 


511 


278604 


14 


47 


668606 


399 


946804 


111 


721702 


510 


278298 


13 


48 


668746 


399 


. 946738 


111 


722009 


510 


277991 


12 


49 


668986 


399 


946671 


111 


722315 


510 


277686 


11 


50 

51 


669225 
9.669464 


399 


946604 


111 
111 


722621 


510 


277379 


10 
9 


398 


9.946638 


9.722927 


510 


10.277073 


52 


669703 


398 


946471 


111 


723232 


509 


276768 


8 


53 


669942 


.398 


946404 


111 


723.';38 


509 


276402 


7 


54 


670181 


.397 


946337 


111 


723844 


509 


2761.56 


6 


•55 


670419 


397 


946270 


112 


724149 


509 


276861 


5 


56 


670668 


397 


946203 


112 


724454 


.509 


276546 


4 


57 


670896 


897 


946136 


112 


724759 


508 


276241 


3 


58 


671134 


396 


946069 


112 


725065 


508 


274935 


2 


59 


671372 


396 


946002 


112 


726369 


508 


274631 


1 


60 


671609 


396 


946936 


112 


726674 


508 


274326 





"^ 


C()i>iiie 


1 


Sine 


Cutaiig. 




1 Taug. 1 M. 1 



62 Degrees. 





46 


(28 DegreeB.J a 


T* 


LB 07 toGABirnino 








~ 


S,r„ 


_»^ 


1 0.«,n,- 1 h 


1 -i'liu^. 


II. 


.:...-.v,^. 


n 












q.iM5113 


nr 


"977a.ro7 


=sr 


TO-^3-33 


fW 












915^H 


u 


735971) 


508 


27403 














915B0 




72G284 




373716 


58 












84573. 


1 


7366S8 


507 


373413 










672SS8 




91666 


1 


7388BS 




S73I08 


66 








a727!l6 




94559 


1 


727J8' 


607 


3728 D3 










673U32 


394 


94553 


I 




507 


S73499 










6T3SS8 




B454M 


1 a 




506 












67360: 


304 


04639 


1 ; 


728 loi 


506 


S7I89I 


52 






■ 


673741 


393 


846338 




728412 


606 


371 588 


51 








673377 


^W 


B463B 
8.945193 




72371B 


508 




60 
49 






B. 8743 13 


9,739030 


505 


10.370980 








8744.13 


393 


946136 


1 ; 


739333 


605 


370677 










674684 


893 


945058 


1 ; 


73963J 


605 


370374 


47 








674910 


333 


9440B0 






605 


S70071 


46 








676155 


383 


944932 


1 i 


73023: 


605 


269767 


45 








875890 


391 


044854 


! ; 


73053; 


505 


269405 


44 








676634 


391 


944786 






504 


369163 










6TB863 


391 


944718 


1 i 


7 1141 


604 


S68S5D 










BT6004 


891 


944650 


1 ; 


7 1444 


604 


368556 











676333 


330 


9445SJ 




7 1746 


604 


368354 


40 






ai 


fi. 6765113 


^iio" 


0.34451. 




9.7 3048 


-5-or 


10.287953 








22 


67079B 


300 


94444f 


4 


7 i351 


503 


267O.10 


38 






23 


677030 


390 


94437- 


4 


7 3653 


603 




37 






34 


677264 


389 


94430[ 


4 


7 3955 


503 




38 






25 


67749S 


369 


944a.l 




7 3257 


503 


S66743 


35 






26 


677731 


389 


B44J7 




7 3558 


503 




34 






37 


677964 


388 


94410 




7 3300 


603 


366140 


33 






at 


678137 




9440.'! 




7 4182 


603 


366M38 


as 






39 


678430 




94396 




734463 


903 




31 






80 


678563 


338 


94383 




734764 


603 




30 






r 


9 6 asoQ 


3 


9 943SJ 


^ 


9 066 


~6or 


"0~a6493: 


89 






as 


6 9 8 


397 


043 6 


4 


T3i19 


502 


364633 


98 






33 


6 0360 




9 1 




06 


50 


864333 


87 






34 


0-9 9 


38 








36403 


36 






35 


a 9834 


38 






50 


S6373 


89 






36 
3 


680058 
680 88 


386 

386 










26343 

36313 


34 
33 






38 


680 9 
68 


3 






6 
600 


S63S3 
2 353 


33 






40 


G3 38 


385 






500 


26233 










il^n 


-386" 


9-94 "T 






500" 


SB 92 








li 


bS 443 


384 


04JU 






500 


B3 








43 


G8 4 


394 


9 30UJ 


6 




499 


SS 33 








44 


68 90 


3S 


9t 93 






499 


3B 03 








45 


683 3 


384 


94^04 




7 92 


499 


6072 








4fi 


68 5 


333 


94. 95 


B 


39 












47 


68 93 


33J 


942 26 


B 


7398 


499 


asoiso 








43 


GS 82 


3S1 


94 6 


8 


40 9 




8 3831 








49 


883 




D 


8 


74046H 


49 










eo 






_5 


40 












r 


rr 






9 OB 
4 3B 


~49F 
498 


6 89.34 
35 635 








63 


6 








498 


363336 








54 








4 98 


40 


8^038 








es 


6 




6 


43 6 


49 


86-7:19 








S6 


6 


3 


9 


559 


49 


2 7441 








e 


6S49 


380 


9420 9 8 


74^ H 


497 










rs 


6 


380 


94 9 B 


i 6 


40 










69 


& 3 


38 ■ 


9 889 


7434 14 


49 










BO 


B 


30 


94 8 9 




496 


256348 








1 II II 


' 


— r=.r, 







nKEB Am TANOENT3. 


(29 Degrees.; 


47 




1 


Stilt 


«. 


rwlrie 1 [>. 1 


T.,., i D. 


C01.„,. 1 








9.BS5&7! 


390 


9.941819 


il7 


9. 7437. W 


496 


10.2563481 


w 










94174 


117 


744050 


496 


355950 


60 










941679 


117 


744348 


496 


265653 


68 






036261 


370 


941609 




744645 


496 


25.53 J5 


67 






636482 


379 


94153 




744943 


496 


255057 


66 






6867U9 


378 


941469 




745240 


496 


2547J0 


55 






G86036 


378 


941398 


117 


745538 


495 


254462 


64 






B87163 


37B 


941328 






495 


264105 


63 






flSTSSa 


378 


94 S58 




746133 




2538G8 


&S 






687616 


377 


94 187 




746429 


495 


263571 


61 






68T843 


377 


94 117 




746726 


406 


253374 


W 






9.eSSU63 




9.94 046 


O 


9.747033 


494 


To:262rn 


40 






6B8SS5 




940975 


118 


747319 


494 


253681 


46 






eset^ii 


376 


940905 




747616 




26238'! 


47 






683747 


376 


940834 




747913 


494 


26208T 


4P 






688972 


376 


9407B3 






494 


251791 


4B 






6S91flS 


376 


940693 


8 


748505 


493 


261495 


44 






689423 


375 


940622 


e 


748801 


493 


351199 


43 






689643 


375 


9405SI 


8 


749097 


493 


350903 


43 




19 


689873 


375 


940480 


8 


749393 


493 


350897 


41 




SO 
21 


6M0U98 


375 
~374" 


940400 


8 

-8 


7496P9 


493 


3.50311 


40 
39 




9.696=33 


9.94U:i38 


9.749935 


-m- 


10.250015 




82 


6gOM8 




940367 


118 


750281 


492 


249719 


S8 




S3 


690773 




946196 


118 


760576 


492 


249434 


87 




34 


690991) 




840125 




750672 


492 


249128 


36 




SS 


601220 


373 


940054 


9 


751167 


492 


2488.1 


36 




SS 


69144* 




939982 


9 


751462 


493 


248.53 


34 




37 


691 S63 


373 


939911 


9 


7S1757 


493 


34824 


S3 




38 


69lflU3 


373 


939840 


9 


7320.12 


491 


24794 


33 




3» 


692115 


372 


939768 


9 


752347 


491 


247653 


31 




30 
8i 


892339 


372 


939697 


9 

119 


763642 


491 


247368 


30 
39 




9.6iJ2l>BS 


~3Tf 


9.930635 


9.752937 


~49r 


1U.S47063 




83 


693785 


37i 


939554 


110 


753331 


491 


246769 


38 




33 


693003 


371 


03918! 


119 


753638 


491 


346474 


27 




84 


6:i3231 


371 


039410 




7.53820 


490 


346180 


28 




3S 


693453 


371 


9393:19 




7.54116 


490 


245885 


28 




36 


693676 


370 


939267 


130 


7iH409 


490 


245S91 






37 


693898 


370 


939195 


130 


764703 


490 


245397 


23 




38 


604120 


370 


939123 


120 


754997 


499 


245003 






39 


694342 


370 


939062 


130 


755391 


490 


244709 


21 




40 






938980 


130 


755585 


489 


244416 


20 




41 


9.694788 


~36y~ 


9.938008 


120 


9.7.56878 


489 


io:3Mr3"2 


19 




42 


095007 


36'! 


93aft}0 


120 


75817 


4S9 


243828 


18 




43 


6952K9 


360 


B387B3 


120 


'5646 


489 


243535 


IT 




44 


09MQU 


3rtB 


0.16691 


130 


76675 


489 


243241 


16 




4B 


B95671 


368 


938619 


ISO 


757053 


4S9 


242948 


16 




46 


BD5892 


368 


938547 


ISO 


757345 


488 


24265 






47 


G961 13 


3C8 


038475 


120 


57638 


488 


242:162 


13 




48 


69na34 


337 


B38403 


121 


75793 


488 


242069 


12 




49 


696554 


367 


938331 


131 


75822 


488 


fi4l776 






60 
51 


696776 


367 
367 


938358 


121 
121 


75851 


488 

-48r 


84 1483 
10.241190 


10 






9.938185 


9.758810 




03 


'697215 


366 


. 938113 


ISl 


759102 


467 


340898 






63 


697435 


366 


538040 121 


750395 


487 


240605 






M 


B97654 




937967 131 


759687 


487 


240313 


6 




69 


697874 


366 


037896 131 


750079 


487 


340031 


6 




S6 


698091 


365 


9978321 131 


760272 


487 


33P73a 


4 




67 


09831 


365 


997749' 181 


780564 


487 


239436 


8 




68 


69S.S3 


365 


937676, 131 


760866 


486 


239144 


2 




60 


69375 


365 


937604' 121 


761148 


486 


238852 


1 




60 


H9897 


301 


9375311 121 


7614.10 


486 


33356 


( 






Coune 




1 «.. 1 


€.«.nR. 





■j "ri:^. \w: 





48 



(30 De^ecs.) a table of LooABiTBinc 



M. I Sine | 1). | dmine 



D. 



Taiig. 



D. 



Coin I in;. 





1 

2 
3 

4 
5 
6 
7 
8 
9 
10 



11 


9.701368 


12 


701585 


13 


701802 


14 


702019 


15 


702230 


16 


702452 


17 


702669 


18 


702885 


19 


703101 


20 


703317 



21 
22 
23 
24 
25 
26 
27 
28 
29 
30 

31 
32 
33 
34 
35 
36 
37 
38 
39 
40 



41 
42 
43 
44 
45 
46 
47 
48 
49 
50 

51 
52 
53 
f.4 
55 
66 
57 
5R 
59 
60 



9.698970 
699189 
699407 
699626 
699844 
700062 
700280 
700498 
700716 
700933 
701151 



9 



.703533 
703749 
703964 
704179 
704395 
704610 
704825 
705040 
705254 
7Q54G9 



9 705683 
705898 
706112 
706326 
706539 
700753 
70G967 
707180 
707393 
707600 



9.707819 
70S032 
70S245 
708458 
708670 
708882 
709094 
709300 
709518 
709730 



9 



709941 
710153 
710364 
710575 
710786 
710997 
711208 
711419 
711629 
711839 



364 
364 
364 
364 
363 
363 
363 
363 
363 
362 
362 



362 
362 
361 
361 
361 
861 
360 
360 
360 
360 



359 
359 
359 
859 
359 
358 
358 
358 
358 
357 



357 
357 
357 
3.58 
356 
356 
356 
355 
355 
355 



355 
3.54 
354 
354 
1354 
353 
353 
353 
353 
353 



352 
352 
352 
352 
351 
.351 
351 
351 
350 
350 



9.937.531 
937458 
937385 
937312 
937238 
937165 
937092 
937019 
936946 
986872 
936799 



9.936725 
936652 
936578 
936505 
9.36431 
936357 
936284 
936210 
936136 
93f)062 



9.935988 
93.5914 
935840 
935766 
935692 
935618 
935543 
935469 
93.5395 
935320 



9.935246 
935171 
935097 
935022 
934948 
934873 
93479S 
93472u 
934649 
934574 



9.934499 
934424 
934349 
934274 
934199 
934123 
934048 
933973 
933898 
933822 



9.933747 
933671 
933596 
933520 
93.3445 
933369 
933293 
933217 
933141 
933066 



121 
122 
122 
122 
122 
122 
122 
122 
122 
122 
122 

122 
123 
123 
123 
123 
123 
123 
123 
123 
123 



123 
123 
123 
124 
124 
124 
124 
124 
124 
124 

124 
124 
124 
124 

124 
124 
125 
125 
125 
125 

126 
]25 
125 
125 
125 
125 
125 
125 
126 
126 

126 
126 
126 
126 
126 
126 
126 
126 
126 
126 



9.761439 


486 


761731 


486 


762023 


486 


762314 


486 


762606 


485 


762897 


485 


763188 


485 


763479 


485 


763770 


485 


764061 


' 485 


764352 


484 


9.764643 


484 


764933 


484 


765224 


484 


765514 


484 


765805 


484 


766095 


484 


766385 


483 


766675 


483 


766965 


483 


767255 


483 


9.767545 


483 


767834 


483 


768124 


482 


768413 


482 


768703 


482 


768992 


482 


769281 


482 


769570 


482 


769860 


481 


770148 


481 


9.770437 


481 


770726 


481 


771015 


481 


771303 


481 


771592 


481 


771880 


480 


772168 


480 


772457 


480 


772745 


480 


773033 


480 


9.773321 


480 


773608 


479 


773896 


479 


774184 


479 


774471 


479 


774759 


479 


77504G 


479 


775333 


479 


775621 


478 


775908 


478 


9.776105 


478 


776482 


478 


776769 


478 


777055 


478 


777342 


478 


777628 


477 


777915 


477 


778201 


477 


778487 


477 


778774 


477 



10.233561160 
238269 
237977 
237686 
237394 
237103 
236812 
836521 
236230 
235939 
235648 



10.235357 
235067 
234776 
234486 
234195 
233906 
233615 
233325 
233036 
232745 



10.232455 
232166 
231876 
231587 
231297 
231008 
230719 
2.30430 
230140 
229852 



10.229563 
229274 
228985 
228697 
228408 
228120 
227832 
227.543 
2272.55 
226967 



10.220679 
226392 
226104 
225816 
225529 
22.5241 
224954 
224667 
224379 
224092 



10.22.3805 
223518 
223231 
222940 
2226.58 
222372 
222085 
221799 
221512 
221226 



} « <^.•^IIIO 



I 



Sli'O 



I I 



(Otillll 



59 i>etiteiw 



I '^^'..K^ I M. I 





miBfl Ajn> tange:cts 


. ^31 De^prees.) 


49 


V. 


1 Bine 


1 D. 


1 CiK^ine t D- 


1 Taiiji. 


1 D. 


1 Vkttnaf. 1 1 





9 711839 


350 


9.933066 


126 


9.778774 


477 


10.221226 


60 


1 


712050 


350 


932990 


127 


779060 


477 


220940 


59 


^ 


712260 


S.'^O 


932914 


187 


779346 


476 


220654 


58 


3 


712469 


349 


932838 


127 


779632 


476 


220368 


67 


4 


712679 


349 


933762 


127 


779918 


476 


220082 


66 


5 


712889 


349 


932685 


127 


780203 


476 


219797 


55 


6 


713098 


349 


932609! 1271 


780489 


476 


219511 


54 


7 


713308 


349 


932533 


127 


780775 


476 


219225 


63 


8 


713517 


348 


9.32457 


127 


781060 


476 


218940 


52 


9 


713726 


348 


932380 


127 


781346 


475 


218654 


51 


10 
11 


713935 
9.714144 


348 


932304 


127 
127 


781631 


475 


218369 


50 
49 


348 


9.932228 


9.781916 


476 


10.218084 


12 

_ 


714352 


.347 


932151 


127 


782201 


475 


217799 


48 


13 


714561 


347 


932075 


128 


732486 


475 


217514 


47 


14 


714769 


347 


93199S 


128 


782771 


475 


217229 


46 


15 


714978 


347 


931921 


128 


783056 


475 


216944 


45 


16 


715186 


347 


931845 


128 


783341 


475 


216659 


44 


17 


715394 


346 


931768 


128 


783626 


474 


216374 


43 


18 


715602 


346 


931691 


128 


783910 


474 


216090 


42 


19 


715809 


346 


931614 


128 


784195 


474 


215805 


41 


20 
21 


716017 


346 


931537 


128 
128 


784479 


474 


215521 


40 
39 


9.716224 


345 


9.931400 


9.784764J 


474 


10.215236 


22 


716432 


315 


931383 


128 


785043 


474 


214952 


38 


23 


716639 


345 


931306 


128 


785332 


473 


214668 


37 


24 


71G846 


345 


931229 


129 


78.5616 


473 


2143vS4 


36 


25 


717053 


345 


931152 


129 


785900 


473 


214100 


35 


26 


717259 


344 


931075 


129 


786184 


473 


213816 


34 


27 


717466 


344 


930998 


129 


785468 


473 


213532 


33 


28 


717673 


344 


930921 


129 


786752 


473 


213248 


32 


29 


717879 


344 


9.30843 


120 


787036 


473 


212964 


31 


30 
31 


718085 


343 


930766 
9.930688 


129 
129 


787319 


472 


212681 


30 
29 


9.718291 


343 


9.787603 


472 


10.212397 


32 


718497 


343 


9306 H 


129 


787880 


472 


212114 


28 


33 


718703 


343 


930533 


129 


788170 


472 


211830 


27 


34 


718909 


343 


930456 


129 


788453 


472 


211547 


26 


35 


719114 


342 


930378 


129 


788736 


472 


211264 


26 


36 


719320 


a42 


930300 


130 


789019 


472 


210981 


24 


37 


719525 


342 


930223 


130 


789302 


471 


210698 


23 


38 


719730 


342 


930145 


130 


'J89585 


471 


210415 


22 


39 


719935 


341 


930067 


130 


789868 


471 


210132 


21 


40 
41 


720140 


341 


929989 


130 
130 


790151 
9.790433 


471 


209849 


20 
19 


9.720345 


341 


9.929911 


471 


10.209567 


42 


720549 


341 


929833 


130 


790716 


471 


209284 


18 


43 


720754 


340 


929755 


130 


790999 


471 


209001 


17 


44 


720958 


340 


929677 


130 


791281 


471 


208719 


16 


45 


721162 


.340 


929599 


130 


791563 


470 


208437 


15 


46 


721366 


340 


929521 


130 


791846 


470 


208154 


14 


47 


721570 


340 


929442 


130 


792128 


470 


207872 


13 


48 


721774 


339 


929364 


131 


792410 


470 


207590 


12 


49 


721978 


339 


929286 


131 


792692 


470 


207308 


11 


50 

51 


722181 


339 


929207 


131 
131 


792974 


470 


207026 


10 
9 


9.722385 


339 


9.929129 


9.793256 


470 


10.206744 


52 


722588 


339 


929050 


131 


793538 


469 


206462 


8 


53 


722791 


338 


928972 


131 


703819 


469 


206181 


7 


54 


722994 


338 


928893 


131 


794101 


469 


205899 


6 


55 


723197 


338 


928815 


131 


794383 


469 


205617 


6 


56 


723400 


338 


928736 


131 


794664 


469 


205336 


4 


57 


723603 


337 


928657 


131 


794945 


469 


205055 


3 


58 


723805 


337 


928^8 


131 


795227 


469 


204773 


2 


59 


724007 


337 


928499 


131 


795508 


468 


204492 


1 


60 


724210 


337 


928420 


131 


795789 


, 468 


204211 





1 


Cot^iiie 




Sim \ 


Coiang. 


1 


Tang. 


r- 



58 Degree!. 



60 


(32 Degrees.) a 


TABLK OF LOOABITHMIO 




M. 


1 Sine 


D. 


1 Cosine | D. 


1 Tunif. 


1 D. 


1 Comnn. 1 1 





9.7-^4210 


337 


9.y28'r^0 


132 


9.795789 


468 


10.204211 


60 


1 


724412 


337 


928342 


132 


796070 


468 


203930 


59 


2 


724614 


336 


928263 


132 


796351 


468 


203649 


58 


3 


724816 


336 


928183 


132 


796632 


468 


203368 


57 


4 


725017 


335 


928104 


132 


796913 


468 


203087 


5f 


5 


725219 


336 


928025 


132 


797194 


468 


202806 


55 


6 


725420 


335 


927946 


132 


797475 


468 


202525 


54 


7 


725622 


335 


927867 


132 


797755 


468 


202245 


53 


8 


725823 


335 


927787 


132 


798036 


467 


201964 


52 


9 


726024 


335 


927708 


132 


798316 


467 


201684 


51 


11 


726226 


335 


927629 


132 
132 


798596 


467 


201404 


50 
49 


9.726426 


334 


9.927649 


9.793877 


467 


10.201123 


12 


726626 


334 


927470 


133 


799157 


467 


200843 


48 


13 


726827 


334 


927390 


133 


799437 


467 


200563 


47 


14 


727027 


334 


927310 


133 


799717 


467 


200283 


46 


15 


727228 


334 


927231 


133 


799997 


466 


200003 


45 


16 


727428 


333 


927151 


133 


800277 


466 


199723 


44 


17 


727628 


333 


927071 


133 


800567 


466 


199443 


43 


18 


727828 


333 


926991 


133 


800836 


466 


199164 


42 


19 


728027 


333 


926911 


133 


801116 


466 


198884 


41 


20 
21 


728227 


333 


926831 


1.33 
133 


801396 


466 


198604 


40 
39 


9.728427 


332 


9.026751 


9.801675 


466 


10.198325 


22 


728626 


332 


926671 


133 


8019,55 


466 


198045 


38 


23 


728825 


332 


926591 


133 


802234 


465 


197766 


37 


24 


729024 


332 


926511 


134 


802513 


465 


197487 


36 


25 


729223 


331 


926431 


134 


802792 


466 


197208 


35 


26 


729422 


.?31 


926361 


134 


803072 


465 


196928 


34 


27 


729621 


331 


920270 


134 


803351 


465 


196649 


33 


28 


729820 


331 


926190 


134 


803630 


465 


196370 


32 


29 


730018 


330 


926110 


134 


803908 


466 


196092 


31 


30 
31 


730216 


330 
330 


926029 


134 
134 


804187 


465 


19.5813 
10.195634 


30 
29 


9.730416 


9.926949 


9.804466 


464 


32 


730613 


330 


925868 


134 


804745 


464 


1962.55 


28 


33 


730811 


330 


925788 


1.34 


80,5023 


464 


194977 


27 


34 


731009 


329 


925707 


134 


805302 


464 


194698 


26 


35 


731206 


329 


92.5626 


134 


806680 


464 


194420 


25 


36 


731404 


329 


925545 


1.35 


805859 


464 


194141 


24 


37 


731602 


329 


925465 


135 


806137 


464 


193863 


23 


38 


731799 


329 


925384 


135 


806415 


463 


193585 


22 


39 


731996 


328 


925303 


135 


806693 


463 


193307 


21 


40 

il 


732193 


328 


925222 
9.925141 


135 
135 


806971 


463 


193029 


20 
19 


9.732390 


328 


9.807249 


463 


10.192751 


42 


732587 


328 


925000 


135 


807527 


463 


192473 


18 


43 


732784 


328 


924979 


135 


807805 


463 


192195 


17 


44 


732980 


327 


924897 


135 


808083 


463 


191917 


16 


45 


733177 


327 


924816 


135 


808361 


463 


191639 


15 


46 


733373 


327 


924735 


136 


80863S 


462 


191362 


14 


47 


733569 


327 


924654 


136 


808916 


462 


191084 


13 


48 


733765 


327 


924572 136 


809193 


462 


190807 12 1 


49 


733961 


326 


924491 


136 


809471 


462 


190.529 


11 


50 
51 


734157 


326 


924409 


136 
136 


809748 


462 


1902,52 


10 
9 


9.734353 


326 


9.924328 


9.810025 


462 


10.189975 


52 


734549 


326 


924246 


136 


810302 


402 


189698 8| 


53 


734744 


325 


924164 


136 


810580 


462 


189420 


7 


54 


734939 


325 


924083 


136 


810857 


462 


189143 


6 


55 


735135 


325 


924001 


136 


811134 


461 


188866 


5 


56 


735330 


325 


923G19 


130 


811410 


461 


188590 


4 


57 


735525 


325 


923^:37 


136 


8116S7 


461 


188313 


3 


58 


735719 


324 


923755 


137 


8r.96i 


461 


188036 


2 


59 


735914 


324 


923373 


137 


812241 


461 


187759 


1 


60 


7:^r>i09 


321 


9'?:vs<M 


137 


8i^\'Sr' 


4R1 


187483 





n 


Cosine 




Siiiu 1 ; 


Cotang. 1 




Tang. 1 M. | 



57 Degrees. 





stsTs AND TA??oE^Ts. (33 DegTces 


'■) 


51 


M. 


1 Sine 


D. 


C'vsire 1 n. 


1 Taiig. 


1 D. 


\' C(>taiis. 1 . 1 





9.7J6109 


324 


9. 92.35.) 1 


137 


9.812517 


461 


10.187482 eol 


1 


736303 


324 


923509 


137 


812794 


461 


187206 


59 


2 


73649S 


324 


923427 


137 


813070 


461 


186930 


58 


3 


736692 


323 


923.345 


137 


813347 


460 


186653 


57 


4 


736886 


323 


923263 


137 


813623 


460 


186377 


56 


5 


737080 


323 


923181 


137 


813899 


460 


186101 


55 


6 


737274 


323 


923098 


137 


814175 


460 


185825 


54 


7 


737467 


323 


923016 


137 


8144.52 


460 


18.5548 


53 


8 


737661 


322 


922933 


137 


814728 


460 


185272 


52 


9 


737855 


322 


922851 


137 


815004 


460 


184996 


51 


10 
11 


738048 


322 


922768 
9.922686 


138 

138 


815279 


460 


184721 


50 
49 


9.738241 


322 


9.815555 


459 


10.184445 


12 


738431 


322 


922603 


138 


815831 


459 


184169 


48 


13 


733627 


321 


922520 


138 


816107 


459 


183893 


47 


14 


738820 


321 


922438 


138 


816.382 


459 


183618 


46 


15 


739013 


.321 


922355 


138 


816658 


459 


183342 


45 


16 


739206 


321 


922272 


138 


816933 


459 


183067 


44 


17 


739398 


321 


922189 


138 


817209 


459 


182791 


43 


18 


739590 


,320 


922106 


138 


817484 


459 


182516 


42 


19 


739783 


320 


922023 


138 


817759 


459 


182241 


41 


20 
21 


739975 
9.740167 


320 


921940 


133 
139 


8180.35 


458 


181965 


40 
39 


320 


9.921857 


9.818310 


458 


10.181690 


22 


740359 


320 


921774 


1.39 


818585 


458 


181415 


38 


23 


740550 


319 


921691 


139 


818860 


458 


181140 


37 


24 


740742 


319 


921607 


1.39 


819135 


458 


180865 


36 


25 


740934 


319 


921.524 


139 


819410 


458 


180590 


35 


26 


741125 


319 


921141 


139 


819684 


458 


180316 


34 


27 


741316 


319 


921357 


139 


819959 


458 


180041 


33 


28 


741508 


318 


921274 


139 


820234 


458 


179766 


32 


29 


741699 


318 


921190 


139 


820508 


457 


179492 


31 


30 
31 


741889 


318 


921107 


139 
139 


820783 


457 


179217 


30 
29 


9.742030 


318 


9.921023 


9.821057 


457 


10.178943 


32 


742271 


318 


920939 


140 


821332 


457 


178668 


28 


33 


742482 


317 


920856 


140 


821606 


457 


178394 


27 


34 


742653 


317 


920772 


140 


821880 


457 


178120 


26 


35 


742842 


317 


920688 


140 


822154 


457 


177846 


25 


36 


743033 


317 


920604 


140 


822429 


457 


177571 


24 


37 


743223 


317 


920520 


140 


82270.3 


457 


177297 


23 


38 


743413 


316 


920436 


140 


822977 


456 


■ 177023 


22 


39 


743602 


316 


920352 


140 


8232.50 


456 


176750 


21 


40 
41 


743792 


316 


920268 
9.920184 


140 
140 


823524 


456 


176476 


20 
19 


9.743982 


316 


9,823798 


456 


I0.17620ar 


42 


744171 


316 


920099 


140 


824072 


456 


175928 


18 


43 


7443^11 


315 


920015 


140 


824345 


456 


175655 


17 


44 


744550 


315 


919931 


141 


824619 


456 


175381 


16 


45 


744739 


315 


919846 


141 


824893 


456 


175107 


15 


46 


744928 


315 


919762 


141 


825166 


456 


174834 


14 


47 


745117 


315 


919677 


141 


825439 


455 


174561 


13 


48 


745308 


314 


919593 


141 


825713 


455 


174287 


12 


49 


745494 


314 


919508 


141 


825986 


455 


174014 


11 


50 
51 


745383 
9.745371 


314 


919424 
9.91.9339 


141 
141 


826259 


455 


173741 


10 
9 


314 


9.826532 


455 


10.173468 


62 


746059 


314 


919254! 141 


826805 


455 


173195 


8 


63 


746248 


313 


919169! 141, 


827078 


455 


172922 


7 


64 


746436 


313 


919085 


141 


827351 


455 


172649 


6 


55 


746624 


313 


919000 


141 


827624 


455 


172376 


6 


56 


746812 


313 


918915 


142 


827897 


454 


172103 


4 


57 


746999 


313 


918830 


142 


823170 


454 


171830 


3 


58 


747187 


312 


918745 


142 


823442 


454 


1715.58 


2 


59 


747374 


312 


9186.59 


142 


828715 


454 


171285 


1 


60 


7475621 


312 


918574 


142 


828987 


454 


171013 






Curiae 



tiitte 



I vJoUllig. I 



Ta-.ir. I M- 



19* 



56 Drtriers. 

G 



cr 



62 


^34 DegreesO a 


TABLE OF LOOARITHSaO 




M. 


Sine 


1 D. 


i CdKine 1 D. 


TaiiR. 


1 D. 


1 iJotnng. j j 





9.747502 


312 


9.918574 


142 


9.828987 


454 


10.171013 


60 


1 


747749 


312 


918489 


142 


829260 


454 


170740 


59 


2 


747936 


312 


918404 


142 


829532 


454 


170468 


68 


3 


748123 


311 


918318 


142 


829305 


454 


170195 


67 


1 


748310 


311 


918233 


142 


830077 


464 


169923 


56 


6 


748497 


311 


918147 


142 


830349 


463 


169651 


55 


6 


748683 


311 


918062 


142 


830621 


453 


169379 


.54 


7 


748870 


311 


917976 


143 


830893 


463 


169107 


53 


a 


749056 


310 


917891 


143 


831165 


453 


168835 


52 


9 


749243 


310 


917805 


143 


831437 


4(f3 


168563 


51 


10 

11 


749429 


310 


917719 


143 
143 


831709 


463 


168291 


60 
49 


9.749615 


310 


9.917634 


9.831981 


463 


10.168019 


12 


749801 


310 


917548 


143 


832253 


453 


167747 


48 


13 


749987 


309 


917462 


143 


832525 


463 


167476 


47 


14 


750172 


309 


917376 


143 


832796 


453 


167204 


46 


15 


750358 


309 


917290 


143 


833068 


452 


166032 


45 


16 


760543 


309 


917204 


143 


833339 


452 


166661 


44 


17 


750729 


309 


917118 


144 


833611 


452 


166389 


43 


18 


750914 


308 


917032 


144 


833882 


452 


166118 


42 


19 


751099 


308 


916946 


144 


834154 


462 


165846 


41 


20 
21 


751384 


308 
308 


916859 
9.916773 


144 
144 


834425 
9.834696 


452 


165575 


40 
39 


9.751469 


452 


10.165304 


22 


751654 


308 


916687 


144 


834967 


452 


165033 


38 


23 


751839 


308 


916600 


144 


835238 


452 


164762 


37 


24 


752023 


307 


916514 


144 


835509 


452 


164491 


.36 


25 


752208 


307 


916427 


144 


836780 


461 


164220 


36 


26 


752392 


307 


916341 


144 


833051 


451 


163949 


34 


27 


752576 


307 


916254 


144 


836322 


461 


163678 


33 


28 


752760 


307 


916167 


145 


836593 


461 


163407 


32 


29 


752944 


306 


916081 


145 


836864 


451 


163136 


31 


30 
31 


753128 


306 


915994 


145 
145 


837134 


451 


162866 


30 
29 


9.753312 


306 


9.915907 


9.837405 


461 


10.162595 


32 


753495 


306 


915820 


146 


837675 


451 


162325 


28 


33 


753679 


306 


916733 


145 


837946 


451 


162054 


27 


34 


753862 


305 


915646 


145 


838216 


451 


161784 


26 


35 


754046 


305 


915559 


145 


838487 


450 


161513 


26 


36 


764229 


305 


915472 


145 


833757 


450 


161243 


24 


37 


754412 


305 


915335 


145 


833027 


450 


160973 


23 


38 


754595 


305 


915297 


145 


839297 


450 


160703 


22 


39 


754778 


304 


915210 


145 


839568 


450 


160432 


21 


40 
41 


7549G0 
9.755143 


304 


915123 
9.915035 


146 
146 


839838 
9.840108 


450 


180162 


20 
19 


304 


450 


10.159802 


42 


755326 


304 


914948 


148 


840378 


450 


159322 


18 


43 


755508 


304 


914860 


146 


840647 


450 


150353 


17 


44 


755690 


304 


914773 


146 


840917 


449 


159083 


16 


45 


755872 


303 


9146S5 


146 


841187 


449 


158813 


16 


46 


756054 


303 


914598 


146 


841457 


449 


158543 


14 


47 


756236 


303 


914510 


146 


841726 


449 


158274 


13 


48 


756418 


303 


914422 


146 


841.996 


449 


169004 


12 


49 


756600 


303 


914334 


146 


842266 


449 


167734 


11 


50 
51 


756782 
9.756963 


302 


914246 
9.914158 


147 
147 


842535 


449 


157465 


10 
9 


302 


9.842805 


449 


10.157195 


52 


757144 


302 


914070 


147 


843074 


449 


156926 


8 


53 


757326 


302 


913982 


147 


843343 


449 


156657 


7 


54 


757507 


302 


913894 


147 


843812 


449 


1.56388 


6 


65 


757688 


301 


913806 


147 


843382 


448 


1.56118 


5 


56 


757869 


301 


913718 


147 


844151 


448 


155849 


4 


57 


758050 


301 


913830 


147 


844420 


448 


155580 


3 


58 


758230 


301 


913541 


147 


8446S9 


448 


155311 


2 


59 


758411 


301 


913463 


147 


844958 


448 


155042 


1 


60 


75S591 


301 


913365 


147 


845227 


44S 


154773 





1 Codiae 


.1 


Sine ) 


Coluiig. 


1 


'J ajiji. 


M 



55 Degrees. 



SOSES AXD TANOEKTs. (35 Dogrees.) 



53 



M. 


1 Fine 


1 D. 


1 Crwne 1 D. 


' Tju«. 


! a 


1 Cn^aic. 1 





9.758591 


301 


9.913365J 147 


9.845227 


443 


i 10.154773 60 


1 


758772 


300 


9132761 147 
913187] 148 


845496 


448 


154504 


69 


2 


758952 


300 


845764 


448 


154236 


58 


3 


759132 


300 


913099! 148 


846033 


448 


163967 


57 


4 


759312 


300 


913010. 148 


846302 


448 


153698! 56 1 


5 


759492 


300 


912922' 148 


846570 


447 


153430 


65 


6 


759672 


299 


912><33 148 


846839 


447 


163161 


54 


7 


759852 


299 


9127H 148 


847107 


447 


162893 


63 


8 


760031 


299 


912655! 148 


847376 


447 


152624 


52 


9 


760211 


299 


912566; 148 


817644 


447 


152356 


51 


10 
11 


760390 


299 


912477 


148 


847913 


447 


152087 


50 


9.760569 


298 


9.912388 


148 


9.848181 


447 


10.151819 49 


12 


760748 


298 


912299 


149 


848449 


447 


151551 48 


13 


760927 


298 


912210 


149 


848717 


447 


161283 47 


14 


761106 


298 


912121 


149 


848986 


447 


151014 46 


15 


761285 


298 


912031 


149 


849254 


447 


150746 45 


16 


761464 


298 


911942 


149 


849522 


447 


150478 


44 


17 


761642 


297 


911853 


149 


849790 


446 


150210 


43 


18 


761821 


297 


911763 


149 


8500.58 


446 


149942 


42 


19 


761999 


297 


911674 


149 


850325 


446 


149675 


41 


20 
21 


762177 


297 


911584 


149 
149 


850593 


446 


149407 


40 
39 


9.762356 


297 


9.911495 


9.850861 


446 


10.149139 


22 


762534 


296 


911405 


149 


851129 


446 


148871 


.38 


23 


762712 


296 


911315 


150 


851396 


U6 


148604 


37 


24 


762889 


296 


911226 


150 


851664 


446 


148336 


36 


26 


763067 


296 


91113G 


150 


851931 


446 


148069 


36 


26 


763245 


296 


911046 


150 


852199 


446 


147801 


34 


27 


763422 


296 


910956 


150 


852466 


446 


147534 


33 


28 


763600 


295 


9lOSf.6 


150 


852733 


445 


147267 


32 


29 


763777 


295 


910776 


150 


853001 


445 


146999 


31 


30 
31 


763954 


295 


9106S6 


150 
150 


, 853268 
9.853535 


445 


146732 


30. 
29 


9.764131 


295 


0.910596 


445 


10.146465 


32 


764308 


295 


910506 


150 


853802 


445 


146198 


28 


33 


764485 


294 


910415 


150 


854069 


445 


145931 


27 


34 


764662 


294 


910325 


151 


854336 


445 


145664 


26 


35 


764838 


294 


910235 


151 


854603 


445 


145397 


25 


36 


765015 


294 


910144 


151 


854870 


445 


145130 


24 


37 


765191 


294 


910054 


151 


855137 


445 


144863 


23 


38 


765367 


294 


909963 


151 


855404 


445 


144.'>96 


22 


39 


765544 


293 


909873 


151 


856671 


444 


144329 


21 


40 
41 


785720 


293 


909782 
9.909691 


151 
151 


855938 


444 


144062 


20 
19 


9.765896 


293 


9.856204 


444 


10.143796 


42 


766072 


293 


909601 


151 


85G471 


444 


143529 


18 


43 


766247 


293 


909510 


151 


856737 


444 


143263 


17 


44 


766423 


293 


909419 


151 


857004 


444 


142996 


16 


45 


766598 


292 


909328 


152 


857270 


444 


142730 


16 


46 


766774 


292 


909237 


152 


857537 


444 


142463 


14 


47 


766949 


292 


909146 


152 


857803 


444 


142197 


13 


48 


767124 


292 


909055 


152 


858069 


444 


141931 


12 


49 


767300 


292 


908964 


152 


858336 


444 


141664 


11 


60 
51 


767475 


291 


908873 
9.908781 


152 
152 


858602 


443 


141398 


10 
'9 


9.767649 


291 


9.858868 


443 


10.141132 


52 


767824 


291 


908690 


152 


859134 


443 


140866 


8 


53 


767999 


291 


908599 


152 


859400 


443 


140600 


7 


54 


768173 


. 291 


908507 


152 


859666 


443 


140334 


6 


55 


768348 


290 


908416 


153 


859932 


443 


140068 


6 


56 


768522 


290 


908324 


153 


860198 


443 


139802 


4 


67 


768697 


290 


908233 


153 


860464 


443 


139536 


3 


58 


768871 


290 


908141 


153 


860730 


443 


139270 


2 


59 


769045 


290 


908049 


153 


860995 


443 


139005 


1 


AO 


769219 


290 


907958' 153 


861261 


443 


138739 







Ot^iiie 1 




I Hiue 1 


1 UolaiiK. 




1 'i-ang. 1 W. 1 



54 Degrees. 



64 


(3C Degrees.] a 


TABI.B OP LOOABITHHIC 














M.| »,w 1 II. 1 roM..^ 1 D. i Ti^nl- 


n. 




^ 






a.Teu-jie 


S»0 


9.9079.i8| 153 


B.861361 


443 


10.138733 






7693 »3 


!89 


007S6H 153 


8R1537 


413 


138473 


68 






76HfiGB 


289 


907774 


63 


861 7D2 


442 


138208 


58 






7eU7AO 


289 


B07082 


S3 


863058 




I37B4: 


£7 






768913 


289 


907600 


S3 








56 






770O«7 


289 


907498 


53 


662589 




137411 


55 






770itiO 


388 


907406 


16. 


8B2854 




13714U 


54 






770433 


388 


B07314 


164 


863119 




136881 


53 






770606 




907333 


154 


863385 




136615 


53 






770779 


388 


907139 


i.'>4 


863650 




136350 


51 






770953 
6.771iaB 


-s- 


907037 


154 


863915 




136085 

lo.iaarfiO 


60 




B 8061)15 


9.HG418U 


443 










80SS52 




86444S 
















80 B 760 


1.S4 


8647111 


443 


13539(1 


47 








287 






864976 


44 




IB 










906575 




865340 


44 




IB 






771087 




906482 


IM 


865505 


44 










778169 


387 


906389 


165 




44 


131230 








772331 


286 


906396 


165 




44 


133965 








778503 


386 


906304 


156 


8HO3O0 


U 


, 133700 








772H7B 


386 


B06111 


166 
155 


866661 


41 




40 
39 




9.773847 


-mT 


9.906018 


9.866830 


44 


10. 3317! 




S2 


773018 


S86 


906925 


155 


867094 


44 




38 




,23 


778190 


S86 


905832 


1.55 


867358 


44 


32B43 


37 




84 


773.131 


S85 


305739 


165 


867633 


41 


33377 


36 




35 


773633 


S95 


905645 


165 


867S87 


44 


3ill3 


36 




36 


773704 


285 


905552 


165 


868153 


440 


3 S48 






B7 


773876 


S85 


B05459 


155 


868416 


440 


3 584 


33 




es 


7741MB 


285 


905360 


156 


868680 


410 


3 331) 


33 




B9 


774317 


88S 


905372 


156 


868945 


440 


a 065 


31 




30 

31 


774388 


284 


905179 


150 
lT6 


869309 




3070 L 
10. 30627 


30 
39 




9.774.^58 


284 


0.806085 


8.869473 


"W 




33 


774739 


284 


904993 


IBB 


869737 


440 


30263 


38 






774899 


384 


904898 


15B 


870001 


410 


39989 


37 




Bl 


775070 


384 


90480-1 


166 


870265 


440 


39735 






S6 


77534U 


384 


904711 


IBS 


870529 


440 


2847] 


35 




88 


77511ft 


383 


904617 


166 


870783 


140 


28307 


34 




37 


775590 


383 


904523 




871057 


110 


128043 


33 




as 


775750 


383 


904439 




871321 


140 


128679 


33 




S9 




283 


901335 




871685 


HI) 


12841.5 


31 




41) 
41 


776001 


283 


B04341 


!57 


871819 




128151 


30 
19 




9.776365 




9.904 147 






10.137888 










904053 




872376 




137624 


18 




43 


7765911 


S83 


903959 




S72640 




127360 


17 




44 


776768 


3^3 


903864 




872B03 


439 


■27097 


16 




4S 


7 7683 1 


383 


B03770 




873167 


439 


136833 


15 




46 




3S3 


903676 




873430 


439 


136570 


11 




47 


77737; 


381 


903591 


57 


873694 


439 


130306 


13 




48 


777444 




8034B7 


57 




439 


136043 






49 


777613 


281 


B03393 


f.8 




439 


1257811 






BO 
51 


777781 


281 


903398 


58 
168 




439 


36516 


10 




0.7779.')0 


381 


9.903203 


9.871747 


139 


10. 3.52.53 




ea 


778118 


3S1 


9U3108 


158 


8760 iO 


139 


340U0 


6 




5a 


7782M7 


380 


903014 


158 






34727 


7 




64 


778455 


380 


903919 


IBS 


875631 




34464 


6 




SB 


77863^ 


380 


902824 


158 


876800 


438 


24300 






5B 




380 


B0272B 


158 


876063 


438 


23937 






57 


778961 


280 


903634 


168 


876326 


438 


33674 






5fl 


779131 


'5" 


B02538 


1B9 


876a«!l 


438 


33111 


3 




6» 






8(12144 


159 


876B51 


438 


33119 






GO 


7794fi; 


279 


802319 


1.59 


877114 


433 


22886 


0^ 




~ 


Uraiii^ 




.1,. 1 


^•■••^;_^ 


i -. («■ 







1 


Hm AlfD TANGEXTS. ^37 1 


l^reei 


^) 


bs 


X. 


Sine 


1 u 1 


C<0ine 1 t>. 


1 Tin* 


1 n. 


1 Cotan*. 1 1 


IT 


9.779463 


279 


9.902319 


159 


9.877114 


438 


10.122386.601 


1 


779f>31 


2^ 


902253 


159 


877377 


438 


122633 


f9 


3 


779798 


379 


902153 


159 


877610 


433 


122360 


Uf 


3 


779936 


379 


902063 


159 


8/7903 


438 


122097 


67 


4 


780133 


379 


901967 


1.59 


878165 


438 


121835 


56 


5 


780300 


278 


901872 


159 


873128 


438 


131.572 


55 


6 


780467 


278 


901776 


159 


878691 


438 


121309 


54 


7 


780634 


27S 


901631 


159 


878953 


437 


121017 


63 


8 


780801 


273 


901535 


159 


879216 


437 


130784 


52 


9 


780958 


278 


901490 


1.59 


879478 


437 


130522 


51 


10 
11 


781131 


378 1 


001394 


160 
160 


879741 
9.830003 


437 


120259 


50 
49 


9.781301 


277 


9.901293 


4:37 


10.119997 


12 


781468 


277 


901202 


160 


880265 


437 


119735 


48 


13 


781634 


277 


901106 


160 


88052-) 


437 


119172 


47 


14 


781800 


277 


901010 


160 


880790 


437 


119210 


46 


15 


781966 


277 


900914 


160 


881052 


437 


1 18948 


45 


16 


782132 


277 


900818 


163 


881314 


437 


118686 


44 


17 


782298 


276 


900722 


160 


881576 


437 


118424 


43 


18 


782464 


276 


900626 


160 


881839 


437 


118161 


42 


19 


782830 


276 


900529 


160 


882101 


437 


117899 


41 


30 
31 


782796 


276 


900433 


161 
161 


832363 


436 


117637 


40 
39 


9.782961 


27a 


9.900337 


9.832625 


436 


10,117375 


32 


733127 


276 


900240 


161 


832837 


436 


117113 


38 


23 


7S3292 


275 


900144 


161 


883148 


436 


116852 


37 


24 


7S34.3S 


275 


900047 


161 


883410 


436 


116590 


36 


25 


783623 


275 


899951 


161 


883672 


436 


116323 


35 


26 


7837SS 


275 


899851 


161 


883934 


436 


116066 


34 


27 


7S3'J5.} 


275 


899757 


161 


884196 


436 


11.5804 


33 


28 


784118 


275 


899660 


161 


834457 


4.36 


115543 


32 


29 


784282 


274 


899564 


161 


881719 


436 


115281 


31 


30 
31 


784447 


274 


899167 
9.899370 


162 
162 


831980 


436 


115020 


30 

29 


9.784612 


2/4 


9.835242 


436 


10.114758 


32 


784776 


274 


899273 


162 


885503 


436 


1 14497 


28 


33 


781911 


274 


899176 


162 


885765 


436 


1 14235 


27 


:I4 


73.') 10') 


274 


899073 


162 


836026 


436 


113974 


26 


35 


785269 


273 


893931 


162 


886288 


436 


113712 


25 


36 


785433 


273 


898334 


162 


886549 


435 


113151 


24 


37 


785597 


273 


893787 


162 


836310 


435 


113190 


23 


38 


785761 


273 


893639 


162 


837072 


435 


112928 


23 


39 


. 785925 


273 


893592 


162 


837333 


435 


112667 


21 


40 
41 


786039 


273 


893194 
9.898397 


163 

163 


837594 
9.837855 


435 
435 


112406 


20 
19 


9.786252 


272 


10.112145 


42 


783416 


272 


898299 


163 


833116 


435 


111884 


18 


43 


786579 


272 


898202 


163 


883377 


435 


111623 


17 


44 


786742 


272 


898104 


163 


883639 


435 


111361 


16 


45 


786906 


272 


898006 


163 


888900 


435 


UllOO 


15 


46 


787069 


272 


897908 


163 


889160 


435 


110840 


14 


47 


787232 


271 


897810 


163 


889421 


1 435 


110579 


13 


48 


787395 


271 


897712 


163 


889632 


435 


110318 


12 


49 


787557 


271 


897614 


163 


889913 


435 


110057 


11 


50 
51 


787720 


271 


897516 
9.897418 


163 
164 


890294 


434 


109796 


10 
9 


9.787883 


271 


9.890465 


434 


10.109535 


52 


788015 


271 


897320 


164 


890725 


434 


109275 


8 


53 


788208 


271 


897222 


164 


890986 


434 


109014 


7 


54 


788370 


270 


897123 


164 


891247 


434 


103753 


6 


55 


788532 


270 


897025 


164 


891.507 


434 


108493 


5 


56 


78S691 


270 


896926 


164 


891768 


434 


108232 


4 


67 


788856 


270 


896823 


164 


892028 


431 


107972 


3 


58 


789018 


270 


896729 


164 


892239 


434 


107711 


2 


59 


789180 


270 


896631 


164 


892519 


434 


107451 


1 


60 


789312 


209 


896532 


164 


892H10 


434 


107190 




Li 


(;«.MiiK 1 




Sine j 


Cotniig. 




1 T^-J*. 










58 DHQr 


enu 








» 






Gf 


1 









66 


(38 Degrees.; a 


TABLE OF LOOABITHinO 




M. 


Sine ] 


0. 


Coaine D. | 


Tnnjj. 


D. 


1 Cotan<;. 


r 





9. 789312 


269 


0.89S53-^ 


164 


9.892810 


HI 


10.107190 


,60 


1 


789504 


289 


896433 


165 


893070 


1069301^91 


2 


78J665 


269 


896335 


165 


893331 


434 


106669 


58 


8 


789S27 


269 


896236 


165 


893591 


434 


106^109 


57 


4 


78998S 


269 


8961.37 


165 


893351 


434 


106149 
1^5889 


56 


* 5 


790149 


269 


896038 


165 


894111 


434 


55 


6 


790310 


268 


895939 


165 


894371 


434 


105629 


54 


7 


. 790471 


263 


895840 


165 


894632 


433 


10.5368 


53 


8 


790632 


268 


895741 


165 


894892 


433 


105108 


52 


9 


790793 


268 


895641 


165 


895152 


433 


104848 


51 


10 
11 


790954 


268 


89.5542 


165 
166 


895412 


433 


104588 60 
10.104333 49 


9.791115 


268 


9.895443 


9.895672 


433 


12 


791275 


267 


895343 


166 


895932 


433 


101068 


48 


13 


791436 


267 


89524^1 


166 


896192 


433 


103808 


47 


14 


791590 


267 


895145 


166 


898452 


433 


103548 


46 


15 


791757 


267 


895045 


166 


896712 


433 


103288 


45 


16 


791917 


267 


894945 


166 


896971 


433 


103029 


44 


17 


792077 


267 


894846 


166 


897231 


433 


102769 


43 


18 


792237 


266 


894746 


166 


897491 


433 


102509 


42 


19 


792397 


266 


894646 


166 


897751 


433 


102249 


41 


20 
21 


79:J557 


266 


894546 


166 
167 


898010 


433 


101990 


40 
39 


9.792716 


266 


9.894446 


9.898270 


433 


10.101730 


22 


792876 


266 


891316 


167 


898530 


433 


101470 


38 


23 


793035 


266 


894246 


167 


898789 


433 


101211 


37 


24 


793195 


265 


894146 


167 


899049 


432 


100951 


36 


25 


793354 


265 


891046 


167 


899308 


432 


100692 


35 


26 


793514 


265 


893946 


167 


899568 


4:)2 


100432 


34 


27 


793673 


265 


89384H 


167 


899827 


432 


100173 


33 


28 


793332 


205 


893745 


167 


900086 


432 


099914 


32 


29 


793991 


265 


893645 


167 


900346 


432 


099654 


31 


30 
31 


791150 
9.791308 


264 


893544 


167 

168 


900605 
9.900864 


432 


099395 


30 
29 


264 


9.893444 


432 


10.099136 


32 


794407 


264 


893343 


168 


901124 


432 


098876 


28 


33 


794626 


264 


893243 


168 


901383 


432 


098617 


27 


34 


791784 


264 


893142 


168 


901642 


432 


098358 


26 


35 


794942 


264 


893041 


168 


901901 


432 


098099 


25 


36 


795101 


264 


892940 


168 


902160 


432 


097840 


24 


37 


795259 


263 


892839 


16S 


902419 


432 


097.581 


23 


38 


795417 


263 


892739 


168 


903679 


432 


097321 


22 


39 


795575 


263 


892638 


168 


902938 


432 


097062 


21 


40 
41 


795733 
9.795891 


263 
263 


892536 
9.892135 


168 
169 


903197 
9.903155 


431 


095803 


20 
19 


431 


10.096545 


42 


796049 


263 


892334 


169 


903714 


431 


09628S 


18 


43 


790206 


263 


892233 


169 


903973 


431 


096027 


17 


44 


796364 


262 


892133 


169 


904232 


431 


095768 


16 


45 


796531 


262 


892030 


169 


904491 


431 


095509 


16 


46 


796G79 


262 


891929 


169 


904750 


431 


095250 


14 


47 


796836 


263 


891827 


169 


905008 


431 


091992 


13 


48 


796993 


262 


891726 


169 


905267 


431 


094733 


12 


49 


797150 


261 


891624 


169 


905526 


431 


094474 


11 


50 

51 


797307 
9.797464 


261 


891523 
9.891421 


170 
170 


905784 
9.900043 


431 


094216 


10 
9 


261 


431 


10.093957 


52 


797621 


261 


891319 


170 


906303 


431 


093698 


8 


53 


797777 


261 


891217 


170 


90C.560 


431 


093440 


7 


54 


797934 


261 


891115 


170 


906819 


431 


093181 


6 


55 


79S09i 


261 


891013 


170 


907077 


431 


092923 


5 


56 


798217 


261 


890911 


170 


907336 


431 


092664 


4 


57 


79^403 


260 


890809 


170 


907594 


431 


092106 


3 


58 


798560 


260 


890707 


170 


907852 


431 


092148 


2 


59 


798716 


260 


89;):)05 


170 


908111 


430 


091889 


1 


60 


79SS72 


260 


899303 


iro 


90S3i)9 


430 


091631 







Cwiiiy 




Sine 1 


Coiaiii. 




'lang. 1 M. 1 



51 l^egrees. 



smss Aio) TANGENTS. (39 Dcgrecs.) 



M. 


1 Sine. 


I I>. 


Cot^itie 1 1). 


1 'J'Riwr- 1 


D. 


Cutang. 1 





9.798872 


260 


9.890503 


17(» 


9.908369 


430 


10.091631 


60 


1 


799028 


260 


890400 


171 


908628 


430 


091372 


59 


2 


799184 


260 


890298 


171 


908886 


430 


091114 


58 


3 


799339 


269 


890195 


171 


909144 


430 


090856 


67 


4 


799496 


259 


890093 


171 


909402 


430 


090598 


56 


5 


799661 


259 


889990 


171 


909660 


430 


090340 


55 


6 


799806 


259 


889888 


171 


909918 


430 


090082 


54 


7 


799962 


269 


889786 


171 


910177 


430 


089823 


53 


8 


800117 


259 


889682 


171 


910435 


430 


089565 


52 


9 


800272 


268 


889679 


171 


910693 


430 


089307 


51 


10 

11 


800427 


268 


889477 
9.889374 


171 
172 


910951 


430 


089049 


50 

49 


9.800582 


268 


9.911209 


430 


10.088791 


12 


800737 


258 


889271 


172 


911.467 


430 


088533 


48 


13 


800892 


258 


889168 


172 


911724 


430 


088276 


47 


14 


801047 


268 


889064 


172 


911982 


430 


088018 


46 


15 


801201 


258 


888961 


172 


912240 


430 


087760 


45 


16 


801356 


257 


888858 


172 


912498 


430 


087502 


44 


17 


801511 


257 


888765 


172 


912756 


430 


087244 


43 


18 


801666 


257 


888661 


172 


913014 


429 


086986 


42 


19 


801819 


257 


888548 


172 


913271 


429 


086729 


41 


20 
21. 


801973 


267 


888444 


173 
173 


913629 


429 


086471 


40 
39 


9.802128 


257 


9.888341 


9.913787 


429 


10.086213 


22 


802282 


256 


888237 


173 


914044 


429 


085956 


38 


23 


' 802136 


256 


888134 


173 


914302 


429 


086698 


37 


24 


802589 


256 


888030 


173 


914660 


429 


086440 


36 


25 


802743 


256 


887926 


173 


914817 


429 


085183 


35 


26 


802897 


256 


887822 


173 


915075 


429 


084925 


34 


27 


803060 


256 


887718 


173 


916332 


429 


084668 


33 


28 


803204 


256 


887614 


173 


915690 


429 


084410 


32 


29 


803367 


256 


887510 


173 


916847 


429 


084153 


31 


30 
31 


803511 
9.803664 


265 


887406 


174 
174 


916104 


429 


083896 


30 
29 


265 


9.887302 


9.916362 


429 


10.083638 


32 


803817 


255 


887198 


174 


916619 


429 


083381 


28 


33 


803970 


265 


887093 


174 


916877 


429 


083123 


2'J 


84 


804123 


255 


886989 


174 


917134 


429 


08286G 


26 


35 


804276 


254 


886885 


174 


917391 


429 


082609 


26 


86 


804428 


254 


886780 


174 


917648 


429 


082352 


24 


37 


804581 


254 


886670 


174 


917905 


429 


082095 


23 


38 


804734 


254 


886671 


174 


918163 


428 


081837 


22 


39 


804886 


264 


886466 


174 


918420 


428 


08'580 


21 


40 
41 


805039 


254 


886362 


175 
176 


.918677 


428 


081323 


20 
19 


9.806191 


254 


9.886267 


9.918934 


428 


10.081066 


42 


805343 


253 


886152 


175 


919191 


428 


080809 


18 


43 


806495 


253 


886047 


176 


919448 


428 


080552 


17 


44 


805647 


253 


885942 


175 


919705 


428 


080295 


16 


45 


805799 


253 


885837 


175 


919962 


428 


080038 


15 


46 


805951 


253 


886732 


175 


920219 


428 


079781 


14 


47 


806103 


253 


886627 


175 


920476 


428 


079624 


13 


48 


806254 


253 


885622 


175 


920733 


428 


079267 


12 


49 


806406 


252 


886416 


176 


920990 


428 


079010 


11 


50 
51 


806557 


262 


885311 


176 
176 


921247 


428 


078753 


10 
9 


9.806709 


252 


9.88.5205 


9.921503 


428 


10.178497 


52 


806860 


252 


885100 


176 


921760 


428 


*»78240 


8 


53 


607011 


252 


884994 


176 


922017 


428 


077983 


7 


54 


807163 


252 


884889 


176 


922274 


428 


07772G 


6 


55 


807314 


262 


884783 


176 


922530 


428 


077470 


5 


56 


807465 


261 


884677 


176 


922787 


428 


077213 


4 


67 


807615 


251 


884572 


176 


923044 


428 


0769.56 


3 


58 


807766 


251 


884466 


176 


923300 


428 


076700 


2 


59 


807917 


251 


884360 


176 


923,557 


427 


076443 


1 


60 


80>?0G7 


251 


884254 


177 


923813 


427 


07618** 






Cosine 




Sine 1 


1 Ootaug. 




Tnng. 





SO Degrees. 



58 


(40 Dcgr 


CCS.") A 


TABLE 3F LdOARITHMXC 




HT 


1 Sine 


1 I>. 


Cot^ine 1 D. 


Taiift. 


1 D. 


Cotai-f!. 1 1 





9.803067 


251 


9.834254! 177 


9.923813 


427 


10.076187 


60 


1 


803318 


251 


884148 


177 


924070 


427 


07593U 


59 


s 


808368 


251 


884042 


177 


924327 


427 


075673 


58 


3 


808519 


250 


8S3936 


177 


924583 


427 


075417 


57 


4 


808669 


250 


883829 


177 


924840 


427 


075160 


56 


5 


8038 19 


250 


883723 


177 


925096 


427 


074904 


55 


6 


808969 


250 


883617 


177 


925352 


427 


074648 


.54 


7 


809119 


250 


883510 


177 


925609 


427 


074391 


53 


8 


809269 


250 


883404 


177 


925865 


427 


074135 


52 


9 


809419 


249 


883297 


178 


926122 


427 


073878 


51 


10 
11 


809569 


249 


883191 


178 
178 


926378 


427 


073622 


50 
49 


9.809718 


249 


9.8S3084 


9.926634 


427 


10.073366 


12 


809868 


249 


832977 


178 


926990 


427 


073110 


48 


13 


810017 


249 


882871 


178 


027147 


427 


072853 


47 


14 


810167 


249 


882764 


178 


927403 


427 


072597 


46 


16 


810316 


248 


882657 


178 


927659 


427 


072341 


45 


16 


810465 


248 


832550 


178 


927915 


427 


072085 


44 


17 


810614 


248 


882443 


178 


928171 


427 


071829 


43 


18 


810763 


248 


882336 


179 


928427 


427 


071573 


42 


19 


810912 


248 


882229 


179 


928683 


427 


071317 


41 


20 
21 


81I0GI 


248 


882121 


179 
179 


928940 


427 


071060 


40 
39 


9.811210 


248 


9.882014 


9.929196 


427 


10.070804 


22 


811358 


247 


881907 


179 


929452 


427 


070548 


38 


23 


811507 


247 


881799 


179 


929708 


427 


070292 


37 


24 


811655 


247 


831692 


179 


929964 


426 


070036 


36 


25 


811804 


247 


881584 


179 


930220 


426 


069780 


35 


26 


811952 


247 


881477 


179 


930475 


426 


069525 


34 


27 


812100 


247 


881369 


179 


939731 


426 


069269 


33 


28 


812248 


247 


881261 


180 


93'J987 


426 


009013 


32 


29 


812396 


246 


881153 


180 


931243 


426 


088757 


31 


30 
31 


812544 
9.812692 


246 


881043 
9.8S093S 


180 
180 


931499 
9.931755 


426 


068501 


30 
29 


246 


426 


10.068245 


82 


812340 


246 


880830 


180 


932010 


426 


087990 


28 


33 


812988 


246 


880722 


180 


932266 


426 


0677.34 


27 


31 


813135 


246 


880613 


180 


932522 


426 


067478 


28 


35 


813233 


246 


8S0505 


ISO 


932778 


420 


067222 


25 


36 


813430 


245 


880397 


180 


933033 


426 


066967 


24 


37 


813578 


245 


8802S9 


181 


933289 


426 


066711 


23 


3S 


813725 


245 


8S0180 


181 


933545 


426 


0664.55 


22 


39 


813872 


245 


830072 


181 


933S00 


426 


066200 


21 


40 
41 


814019 
9.814166 


245 


87996^3 
9.879855 


181 
181 


934056 
9.934311 


426 


065944 


20 
19 


245 


426 


10.065689 


42 


814313 


245 


879746 


i8i 


934537 


426 


065433 


18 


43 


814460 


244 


879337 


181 


934S23 


426 


065177 


17 


44 


814607 


244 


879529 


181 


935378 


426 


0G4922 


16 


45 


814; 53 


241 


879420 


181 


935333 


426 


064667 


15 


46 


814900 


244 


879311 


181 


935589 


426 


06441 1 


14 


47 


815046 


244 


879232| 182 


935844 


426 


064156 


13 


48 


815193 


244 


879093 


182 


936100 


426 


063900 


12 


49 


81533'J 


244 


8789S4 


182 


936355 


426 


063345 


11 


50 
51 


815485 
9.815631 


243 


87SJ75 


182 


936610 
9.936866 


426 


063390 


10 
9 


243 


9.878766! 182 


425 


10.0631:34 


53 


815773 


243 


8786581 182 


937121 


425 


062S79 


8 


53 


815924 


243 


878547i 182 


937376 


425 


062624 


7 


54 


816069 


243 


8784.3S. 182 


937632 


425 


062368 


6 


55 


816215 


243 


87S32S 182 


9378S7 


425 


062113 


5 


56 


8163'>l 


; 243 


878219 183 


938 142 


425 


061858 


4 


57 


816507 


] 242 


878109, 183 


93^398 


425 


061602 


3 


58 


816^)52 


242 


877999, 183 


93S653 


425 


031347 


2 


59 


813798 


' 242 


877890' 183 


93.-193^ 


425 


061002 


1 


60 


816943 


' 242 


87778) is:j 


933163 


425 


060S37 







1 Cosine 




Biix^ 1 


j roia/iy. 


1 


•I'uiig. 1 M. 1 








49 


Disgr 


ees 









snrcs Ain> TAKOENTB. ^41 Degrees.) 



59 



M. 


1 Sine 


D. 


Ci>Riiie 1 D. 


Tnnir. 


D. 


Cotang. 1 1 





9.816943 


242 


9.877780 


183 


9.939163 


426 


10.060837 


60 


1 


817088 


242 


877670 


163 


939418 


426 


060582 


69 


2 


817233 


242 


877560 


183 


939673 


426 


060327 


68 


» 


817379 


242 


877460 


1S3 


939928 


425 


060072 


67 


4 


817524 


241 


877340 


183 


940183 


425 


059817 


56 


5 


817668 


241 


877230 


184 


940438 


426 


05J562 


56 


6 


817813 


241 


877120 


184 


940594 


426 


059306 


54 


7 


817958 


241 


877010 


184 


940949 


426 


059051 


53 


8 


818103 


241 


876899 


184 


941204 


425 


068796 


62 


9 


818247 


241 


876789 


184 


941458 


425 


06S542 


51 


10 
11 


818392 


241 


876678 


184 
184 


941714 


426 


068286 


60 
49 


9.818536 


240 


9.876568 


9.941968 


425 


10.058032 


12 


81868] 


240 


876457 


184 


942223 


426 


057777 


48 


18 


818825 


240 


876347 


184 


942478 


426 


067522 


47 


14 


818969 


240 


876236 


185 


942733 


425 


067267 


46 


16 


819113 


240 


876125 


185 


942988 


426 


057012 


45 


16 


819257 


240 


876014 


185 


943243 


426 


056757 


44 


17 


819401 


240 


875904 


185 


943498 


426 


056502 


43 


18 


819545 


239 


875793 


186 


943752 


426 


056248 


42 


19 


819689 


239 


875682 


^86 


944007 


426 


055993 


41 


20 
21 


819832 


239 


875571 


185 
185 


944262 


426 


0537.38 


40 
39 


9.819976 


239 


9.875459 


9.944517 


425 


10.05.5483 


22 


820120 


239 


875348 


185 


944771 


424 


055229 


38 


23 


820263 


239 


875237 


185 


945026 


424 


054974 


37 


24 


820406 


239 


875126 


186 


945281 


424 


054719 


36 


26 


820550 


238 


875014 


186 


946535 


424 


054465 


36 


26 


820693 


238 


874903 


186 


945790 


424 


054210 


34 


27 


820836 


238 


874791 


186 


946045 


424 


. 053955 


33 


28 


820979 


238 


874680 


186 


946299 


424 


0.53701 


32 


29 


821122 


238 


874568 


186 


946554 


424 


053446 


31 


30 
31 


821265 


238 


874456 


180 
186 


946808 


424 


053192 


30 

29 


9.821407 


238 


9.874344 


9.947063 


424 


10.052937 


82 


821550 


238 


874232 


187 


• 947318 


424 


052682 


28 


33 


821693 


237 


874121 


187 


947572 


424 


052428 


27 


34 


821835 


237 


874009 


187 


947826 


424 


052174 


26 


36 


821977 


237 


873896 


187 


948081 


424 


051919 


25 


86 


822120 


237 


873784 


187 


948336 


424 


051664 


24 


87 


822262 


237 


873672 


187 


948590 


424 


051410 


23 


38 


822404 


237 


873560 


187 


948844 


424 


051156 


22 


39 


822546 


237 


873448 


187 


949099 


424 


050901 


21 


40 

41 


822688 


236 


873335 


187 
187 


949353 


424 


050647 


20 
19 


9.822830 


236 


9.873223 


9.949607 


424 


10.050393 


42 


822972 


236 


873110 


188 


949862 


424 


050138 


18 


43 


823114 


236 


872998 


188 


9.50116 


424 


049884 


17 


44 


823255 


236 


872885 


188 


9,50370 


424 


049630 


16 


45 


823397 


236 


872772 


188 


950625 


424 


049375 


16 


46 


823539 


236 


872659 


188 


9.50879 


424 


049121 


14 


47 


823680 


235 


872547 


188 


951133 


424 


048867 


13 


48 


823821 


235 


872434 


188 


951388 


424 


048612 


12 


49 


823963 


235 


872321 


188 


951642 


424 


048358 


11 


60 
61 


824104 


235 


872208 
9.872095 


188 
189 


951896 


424 


048104 


10 
9 


9.824245 


235 


9.1>52150 


424 


10.047850 


62 


824386 


235 


871981 


189 


952405 


424 


047695 


8 


63 


824527 


235 


871868 


189 


952659 


424 


047341 


7 


64 


824668 


234 


871755 


189 


952913 


424 


047087 


6 


65 


824808 


234 


871641 


189 


953167 


423 


046833 


6 


66 


824949 


234 


871528 


189 


953421 


423 


046579 


4 


67 


825090 


234 


871414 


189 


963675 


423 


046325 


3 


68 


825230 


234 


871.301 


189 


953929 


423 


046071 


2 


69 


825371 


234 


871187 


189 


954183 


423 


045817 


1 


60 


825611 


234 


871073 


190 


954437 


423 


04.5563 







1 Cosine 


! 


dine 1 1 


Colaitg. 


1 


Tang. 1 M. 1 






20 


48 


Degm 


99. 











eo 


(42 Degrees.) a 


TABLK OF LOOABTTHMIC 




• 


M. 


1 Sine 


D. 


1 Cwino 1 D. 


Tanc. 


D. 


1 Cotnn{!. 









9.826611 


234 


9.871073 


190 


9.954437 


423 


10.045663 


60 




1 


826661 


233 


870960 


190 


964691 


423 


046.309 


69 




2 


826791 


233 


870846 


190 


964945 


423 


046056 


58 




3 


826931 


233 


870732 


190 


966200 


423 


044800 


67 


1 

1 


4 


826071 


233 


870618 


190 


966464 


423 


044646 


66 


! 
1 


5 


826211 


233 


870504 


190 


966707 


423 


044293 


66 




6 


826361 


233 


870390 


190 


966961 


423 


044039 


54 




7 


826491 


233 


870276 


190 


966216 


423 


043785 


63 




8 


826631 


233 


870161 


190 


966469 


423 


043631 


62 




9 


826770 


232 


870047 


191 


966723 


423 


043277 


61 


' 


10 
11 


826910 


232 


869933 


191 
191 


966977 


423 


043023 


60 
49 


1 


9.827049 


232 


9.869818 


9.967231 


423 


10.042769 




12 


827189 


232 


869704 


191 


967486 


423 


042515 


48 




13 


827328 


232 


869689 


191 


967739 


423 


042261 


47 




14 


827467 


232 


869474 


191 


967993 


423 


042007 


46 




16 


827606 


232 


869360 


191 


968246 


423 


041754 


45 




16 


827746 


232 


869246 


191 


.958600 


423 


041500 


44 




17 


827884 


231 


8691.30 


191 


958764 


423 


041246 


43 




18 


82S023 


231 


869016 


192 


969008 


423 


040992 


42 




19 


828162 


231 


868900 


192 


959262 


423 


040738 


41 




20 
21 


828301 


231 


868785 


192 
192 


959516 


423 


040484 


40 
39 




9.828439 


231 


9.868670 


9.959769 


423 


10.040231 




22 


828578 


231 


868655 


192 


960023 


423 


039977 


38 




23 


828716 


23] 


868440 


192 


960277 


423 


039723 


37 




24 


828856 


230 


868324 


192 


9G0531 


423 


039469 


36 




25 


828993 


230 


868209 


192 


960784 


423 


039216 


35 




2G 


829131 


230 


868093 


192 


961038 


428 


038962 


34 




27 


829269 


230 


867978 


193 


961291 


423 


038709 


33 




28 


829407 


230 


867862 


193 


961645 


423 


038466 


32 




29 


829645 


230 


867747 


193 


961799 


423 


0.38201 


31 




30 
31 


829683 


230 


867631 


193 
193 


962052 


423 


037948 


30 
29 




9.829821 


229 


9.867516 


9.962306 


423 


10.037694 




32 


829969 


229 


867399 


193 


962560 


423 


037440 


28 




33 


830097 


229 


867283 


1^3 


962813 


423 


037187 


27 




34 


8302.34 


229 


867167 


193 


963067 


423 


036933 


26 




35 


830372 


229 


867061 


193 


963320 


423 


036680 


25 




36 


830509 


229 


866935 


194 


963574 


423 


U36426 


24 




37 


830646 


229 


866819 


194 


963827 


423 


036173 


23 




38 


8307M 


229 


866703 


194 


964081 


423 


035919 


22 




39 


830921 


228 


866586 


194 


964335 


423 


035666 


21 




40 
41 


831058 


228 


866470 


194 
194 


964588 


422 


035412 


20 
19 




9.831195 


228 


9.866353 


9.964842 


422 • 


10.035158 




42 


831332 


228 


866237 


194 


965095 


422 


034906 


18 




43 


831469 


228 


866120 


194 


965349 


422 


034651 


17 




44 


831606 


228 


866004 


195 


965602 


422 


034398 


16 




46 


831742 


228 


865887 


195 


965855 


422 


034146 


15 




46 


831879 


228 


8G5770 


196 


966109 


422 


033891 


14 




47 


832015 


227 


865653 


195 


966362 


422 


0336.38 


13 




48 


832152 


227 


865636 


195 


966616 


422 


033384 


12 




49 


832288 


227 


865419 


195 


966869 


422 


033131 


11 




60 
51 


832425 


227 


865302 


195 
195 


967123 


422 


032877 


10 
9 




9.832561 


227 


9.865185 


9.967376 


422 


10.032624 




62 


832697 


227 


865068 


195 


967629 


422 


032371 


8 




53 


832833 


227 


864950 


195 


967883 


422 


032117 


7 




54 


832969 


226 


864833 


196 


968136 


422 


031864 


6 




65 


833105 


226 


864716 196 


968389 


422 


031611 


6 




56 


833241 


226 


864598 


196 


968643 


422 


0313.57 


4 




57 


833377 


226 


864481 


196 


968896 


422 


031104 


3 




58 


833512 


226 


864363 


196 


969149 


422 


030851 


2 




59 


833648 


226 


864246 


196 


969403 


422 


030697 


1 




60 


833783 


226 


864127* 196' 


969656 


422 


030344 





* 




Cosiiiu 1 




1 Sine 1 1 


Cotang. 1 




Tang. 1 M. | 



47 Degrees. 





BIlfBS AND TANGE19TS 


u (43 De^irrees ) 


61 


« 


M. 


Sine 1 


D. 


1 Cosine | D. 


Tani{. 


1 D. 


Cotang. 1 







9.833783 


226 


9.864127 


196)9.969656 


422 


10.0303441 60 




1 


833919 


225 


864010 


196 


969909 


422 


030091 


59 




2 


834054 


225 


863892 


197 


970162 


422 


029838 


58 




3 


834189 


225 


863774 


197 


970416 


422 


029584 


57 




4 


834325 


225 


863656 


197 


970669 


422 


029331 


56 




6 


834460 


225 


863538 


197 


970922 


422 


029078 


.55 


» 


6 


834596 


225 


863419 


197 


971175 


422 


028825 


54 


1 


7 


834730 


225 


863301 


197 


971429 


422 


028571 


53 


1 


8 


834866 


225 


863183 


197 


971682 


422 


028318 


52 


» 


9 


834999 


224 


863064 


197 


971935 


422 


028066 


51 


* 


10 
11 


835134 


224 


862946 


198 
198 


972188 


422 


027812 


50 
49 


; 


9.835269 


224 


9.862827 


9.972441 


422 


10.027559 


1 


12 


835403 


224 


862709 


198 


972694 


422 


027306 


48 


1 


13 


835538 


224 


862590 


198 


972948 


422 


027052 


47 




14 


835672 


224 


862471 


198 


973201 


422 


026799 


46 




16 


835807 


224 


862353 


198 


973454 


422 


026546 


45 




16 


835941 


224 


862234 


198 


973707 


422 


026293 


44 




17 


836075 


223 


862115 


198 


973960 


422 


026040 


43 




18 


836209 


223 


861996 


198 


974213 


422 


026787 


42 




19 


836343 


223 


861877 


198 


974466 


422 


025534 


41 




20 
21 


836477 
9.836611 


223 


861758 


199 
199 


974719 


422 


025281 


40 
§9 




223 


9.861638 


9.974973 


422 


10.02.5027 




22 


836745 


223 


861519 


199 


975226 


422 


024774 


.38 




23 


836878 


223 


861400 


199 


975479 


422 


024521 


37 




24 


837012 


222 


861280 


199 


975732 


422 


024268 


36 




25 


837146 


222 


861161 


199 


975985 


422 


024015 


35 




26 


837279 


222 


861041 


199 


976238 


422 


023762 


34 




27 


837412 


222 


860922 


199 


976491 


422 


023509 


33 




28 


837546 


222 


860802 


199 


976744 


422 


023256 


32 




29 


837679 


222 


860682 


200 


976997 


422 


023003 


31 




30 
31 


837812 


222 


860562 


200 
200 


977250 


422 


022750 


30 
29 




9.837945 


222 


9.860442 


9.977603 


422 


10.022497 




32 


838078 


221 


860322 


200 


977756 


422 


022244 


28 




38 


838211 


221 


860202 


200 


978009 


422 


021991 


27 




34 


838344 


221 


860082 


200 


978262 


422 


021738 


26 




35 


838477 


221 


859962 


200 


978515 


422 


021485 


25 




36 


838610 


221 


859842 


200 


978768 


422 


021232 


24 




37 


838742 


221 


859721 


201 


979021 


422 


020979 


23 




38 


838875 


221 


859601 


201 


979274 


422 


020726 


22 




39 


839007 


221 


859480 


201 


979527 


422 


020473 


21 




40 
41 


839140 


220 


859360 


201 
201 


979780 


422 


020220 


20 
19 




9.839272 


220 


9.869239 


9.980033 


422 


10.019967 




42 


839404 


220 


859119 


201 


980286 


422 


019714 


18 




43 


839536 


220 


858998 


201 


980.538 


422 


019462 


17 




44 


839668 


220 


858877 


201 


980791 


421 


019209 


10 




45 


839800 


220 


858756 


202 


981044 


421 


018956 


15 




46 


839932 


220 


858635 


202 


981297 


421 


018703 


14 




47 


840064 


219 


858514 


202 


981550 


421 


018460 


13 




48 


840196 


219 


858393 


202 


981803 


421 


018197 


12 




49 


840328 


219 


858272 


202 


982056 


421 


017944 


11 




50 
51 


840459 


219 


858151 
9.858029 


202 
202 


982309 


421 


017691 


10 
9 




9.840591 


219 


9.982562 


421 


10.017438 




52 


840722 


219 


857908 


202 


982814 


421 


017186 


8 


1 


53 


840854 


219 


857786 


202 


983067 


421 


016933 


7 


1 


54 


840986 


219 


857665 


203 


983320 


421 


016680 


6 


1 


55 


841116 


218 


857543 


203 


983573 


421 


016427 


5 


1 

1 


56 


841247 


218 


857422 


203 


983826 


421 


016174 


4 




57 


841378 


218 


857300 


203 


984079 


421 


015921 


3 


( 
1 


58 


841509 


218 


857178 


203 


984331 


421 


015669 2 


1 


59 


841640 


218 


857056 


203 


984584 


421 


015416 1 




60 


841771 


218 


856934< 


f203 


984837 


421 


015163 


1 




1 Cowne 




1 Sine 1 1 


Cotang. 


1 


1 'Jang. 1 M. 












46 


Degrees. 











62 


(44 Degr 


ees.) A 


TABLB OF LOGAJOTHiaO 




n.l 


Sine 


D. 1 


Coiiine | Ji. 


. Taiiff. 


1 D. 


I Cotanff. 1 





9.841771 


218 


9.8569341203 


9.984837 


421 


10.015163 


60 


1 


841902 


218 


856812 


203 


985090 


421 


014910 


59 


2 


842033 


218 


856690 


204 


985343 


421 


014657 


58 


3 


842163 


217 


856568 


204 


985596 


421 


014404 


57 


4 


842294 


217 


856446 


204 


985848 


421 


0141.52 


56 


6 


842424 


217 


856323 


204 


986101 


421 


013899 


55 


6 


842555 


217 


856201 


204 


986354 


421 


013646 


.54 


7 


842685 


217 


856078 


204 


986607 


421 


013393 


53 


8 


842815 


217 


855956 


204 


986860 


421 


013140 


52 


9 


842946 


217 


855833 


204 


987112 


421 


012888 


51 


10 
11 


843076 


217 


855711 


205 
205 


987365 


421 


012635 


50 
49 


9.843206 


2l6 


9.855588 


9.987618 


421 


10.012.382 


12 


843336 


216 


855465 


205 


987871 


421 


012129 


48 


13 


843466 


216 


855342 


205 


988123 


421 


011877 


47 


14 


843595 


216 


855219 


205 


988376 


421 


011624 


46 


15 


843725 


216 


855096 


205 


988629 


421 


011371 


45 


16 


843855 


216 


854973 


205 


988882 


421 


011118 


44 


17 


843984 


216 


854850 


205 


989134 


421 


010866 


43 


18 


844114 


215 


854727 


206 


989387 


421 


010613 


42 


19 


844243 


215 


854603 


206 


989640 


421 


010360 


41 


20 
21 


8^M372 


215 


854180 


206 
206 


989893 


421 


010107 


40 
30 


9.844502 


2l5 


9.854:356 


9.090145 


421 


10.009855 


22 


844631 


215 


854233 


206 


990398 


421 


009602 


38 


23 


844760 


215 


854109 


206 


990651 


421 


009349 


37 


24 


844889 


215 


853986 


206 


990903 


421 


009097 


36 


25 


845018 


215 


853862 


206 


991156 


421 


008844 


85 


26 


845147 


215 


853738 


206 


991409 


421 


008591 


34 


27 


845276 


214 


853614 


207 


991662 


421 


008338 


33 


28 


845405 


214 


853490 


207 


991914 


421 


008086 


32 


23 


a45533 


214 


853366 


207 


992167 


421 


007833 


31 


30 
31 


845662 


214 


853242 


207 
207 


992420 


421 


007580 


30 
29 


9.845790 


214 


9.853118 


9.992672 


421 


10 007328 


32 


845919 


214 


852994 


207 


992925 


421 


007075 


28 


33 


846047 


214 


8.52869 


207 


993178 


421 


006822 


27 


34 


846175 


214 


852745 


207 


993430 


421 


006570 


26 


35 


846304 


214 


852620 


207 


993683 


421 


006317 


25 


36 


846432 


213 


852496 


208 


993936 


421 


006064 


24 


37 


846560 


213 


852371 


208 


994189 


421 


00.5811 


23 


38 


846688 


213 


852247 


208 


994441 


42i 


00.5559 


22 


39 


8408 16 


213 


852122 


208 


994694 


421 


005306 


21 


40 
41 


846944 
9.847071 


213 


851997 


208 
208 


994947 
9.995199 


421 


005053 


20 
19 


213 


9.851872 


421 


10.004801 


42 


84719!) 


213 


851747 


208 


995452 


421 


004548 


18 


43 


847327 


213 


851622 


208 


995705 


421 


004295 


17 


44 


847454 


212 


851497 


209 


995957 


421 


004043 


16 


45 


8475S2 


212 


851372 


209 


990210 


421 


033790 


15 


46 


847709 


212 


851246 


209 


99S463 


421 


003537 


14 


47 


847836 


212 


851121 


209 


996715 


421 


003285 


13 


48 


8479B4 


212 


850996 


209 


996968 


421 


003032 


12 


49 


848091 


212 


850870 


209 


907221 


421 


002779 


11 


50 
51 


848218 
9.848345 


212 


850745 


209 
209 


997473 
9.997726 


421 


002527 


10 
"9 


212 


9.850619 


421 


10.002274 


52 


848472 


211 


850493 


210 


997979 


421 


002021 


8 


53 


848599 


211 


850368 


210 


998231 


421 


001709 


7 


54 


84872G 


211 


850242 


210 


998484 


421 


001516 


6 


55 


848852 


211 


850116 


210 


998737 


421 


001263 


5 


56 


848979 


211 


849990 


210 


99S989 


421 


001011 


4 


57 


849106 


211 


849S64 


210 


999242 


421 


000758 


3 


58 


840232 


211 


849738 


210 


999495 


421 


000505 


2 


59 


849359 


211 


849611 


210 


• 999748 


421 


000253 1 
000000 


60 


819185 


211 


8494S5 210 


10.000000 


421 


1 


Cojiine 




]EJine 


Co* ail?. 


1 


Taus. 1 M. 



45 Degrees. 



A TABLE OF UTATUBAL SI\CS. 

I U »eg- I I Dee- I ''"ag- | a Dug. Ir 40^ 



H| 



jasa ,|U 

U|9999 

99 a u 



J flMUa^ l u 46 sm4l) UaJOS MiliJM ||l] 
M N 0- 1 Nu. I N ji- 1 Nut f 

rOoB. ii B8 Dag. |l 87 Doe, II" 



"Dag, i 87 Peg. I B8 



t itATcxAii amxs. 



yi 


h l).e- 


|,_66«,. 


■ 7D.,, 


8D.g 


UDee 












N8 




NB INCa 




1 


4 9G 
(J 4 6 J 

OHH S 60S 


ii i: 


» 


90 5 
99348 
9 ii 

93240 


39 7 
394b 
3975 
4004 
4033 

or 


^921 
190 


6 3 9B G9 

1 G 4 
5 1 

1 


9 

5. 


20 


n 

09 3 

09 H2 
0^4 




UO . 9 J 

* 






J 


- 








-; 


Zi 9 
J5 1 








.. 


e 










3 H 90 
G B3 9B58 
B 9Ba8( 

i 


so 

9 

a 


441 lit 
46 a 

46l ~^l 
J 1 


— 










I»— 1 




l^ir 


1 
1 


¥ 


sr^^Ts- 




— lirs^ 


M 












B 


80D,g 





A TABLE OF NATUSAL nKBS. 



M U=g. II 14 Ubi. 



ee i9(i5a | flai6B |2076ji 



2076ll97fiil |,i2lfi7 97444 iMlSl^g 

»-. y|— 8 nt^InTbt -ash 

7H l)ng. II 77 Uotr. I 78 IH 



I iiH 1)7058 
0nlfl7061 
;)fi|37044 
"97037 



2.^7l3!96B3a ( 

25741196831) 1 

a576fl'.UBaz3 ■ 

25708 :gu6 15 : 

25854 96600 



75 ll«g. 



A TABLE 07 NATTTKAL OSm. 







lOU-g. 1 J7 1ie,;. lU Ueg. |, 19 beg. 








N.S .I^ l^H 






- 


asS52 


9fi593 


a75rtj!9r,ia(i syj:i7b;.(i3>i :(ii'Ji)j!tst.iJ'-j^5,i7|!M553 


^ 






S59IU 


96.1H5 




S9 






sr.938 


9rifl7B 


■;,,,■;. ■1..!,^ ■:■(:..,.■...-.■. I.M..IU,...,, '^-ll;!^! 


BB 






ssaaa 


)657l)i|u^ ■ .1 . '..■..!-,■ -■ |,| '11523 


57 






fibasi 




;-,■.. ■ ■'. .■■■.■,; ... :ii,',i4 


66 






ZGOia 




■j;,,. ...,,,, , ,Mr,o^ 








aiiti.-,<i 1-.;^ 




M 








. - , ' ■'■ 1 Ij-i4^; 


53 








-■ ., , ■■1475 


52 








■ ■ . . .; :iU6( 


51 










50 






BliliJi 


-.-"i^iui^ 


49 






sii^n 


-;i'.iii3s 


la 






aii-j.i . 


ii'ii.ua 








Et;^:,.. 


"JlUS 


4G 






aiiau. 


■ ■ ■ ■ . . . i.ji.iuo 


4G 








■:';>i;t3H 


u 








,, . ... . . :';,l;)90 


13 








.' 'i-iaao 


12 






Stiil.-. ■ ■ 


... . '. ;it:i70 








aiii 1 : . 


■■.■,;H:iSl 


4lt 






2fi.l,l ■■ . 


... • ;l :il;l31 


39 






2iir,(i;' 


.S13.I! 


38 




8t 


2H:r, - 




87 




34 


MIl.-i.Ti. ■ 




3e 




S5 


ac.-i.^i : ■■■ 




111313 


35 




86 






;J!303 


34 




87 






i:(aMa 


U-1293 


33 




S* 








942S4 


32 








■ ''■'■'^ 


33353 


94874 


31 




30 






3339] 


94aS4 


30 




Si 






VAUH 


9mS 


29 




38 


3r,7BrM;<ai7,a^4r>7|!).>-.:.-. ^ . ... , -rrv 




28 




33 


25808 


96310 


■Mwsa.wr.; .... ■ . ../iiass 


27 




34 


26S36 


nnraa 


*85l3;)f.SJ[) :(..:■ . ..■■icjas 


26 




US!! 


9g:ui 
yR3ir 


[285.11 MSII .(..; ■ -n-zu 


23 




37 


3B9SI} 


yeaos 


;■-**.'''■ '''-"■-"■ .' 'ui9f 


33 








yaaiH 


.■;«.-. . ..- . ,.i;[,.ii8t, 


22 




41 


aTDO'l 


Lifiswii 


■>'■-. -. :in7C 
-•''.■ ■'ill67 


21 
20 






■i70S-> i 11127 7 


■■:■■.' . ■ - (1157 
■ ! :, !,U37 






4: 

41 


a7D(!i)'i";.^-ii 


S 




4! 


a(|.i', . : ■ . ■■ \:',\\% 


5 




4« 
47 


aJiou -■ ..■',.' ■ ■■■'■Mioi 


"4 




4a 
4a 


V^j' /'.■'■:' '''^'■'''«' 


la 




a 


aj;;^; . ...,,,()8( 


10 




i,':\\- ■■ ; -'j-iosa 


9 




5; 


27. .- ■■ ...-■,, ■ , ,,11,39 


7 




65 


o'''i ''■' ,-, ' ^ ''"'■■^- 


6 




6?.i;i::,> .::;-.. ,.,...;,. ,.-....,-,:.; ... -,:;- . ,.;.;,;^.'"?^ 


4 
3 




fi8I^7so^■>.,N;,■;^'|'■■^|y.'>"1. ^^-i-!!;.!^! ;!'^:.ii^|:Mr,7i|i..ii.n'9.m9 


a 




Bfl,s753F;|y(.iyi|,^d-;iJuu5i;3u,,au874|9snsaaaiui)ifitii 14175,03979 






H 


N.w, N.s. Ni^i N8. srcSTi-JTsT tM'«:lpor NTwrlNTsT 


s: 






74D^jl, 11 73D«g. II 7a Deft. II 71 Dog.i 70 Deg. 





L TABLE OF ItATVKAL ■ 





SU^._^, E 1) 






K 


S4i)., ■ 





H 


^F|^*UlF 




M 




26B3 9_M 


9 9 a 


A 




64 3U GO 


S 9 408GO 9 3 3 


i 




6 63 3 28 


9 959 40880 9 360 






6!0 3B3 4 


S 948 409 3t9 34« 






609 3 34 
598, 3938 


S 93G 1 939 
» 9e5 40966 




49 




j8 39394 


9 4 40993 




48 




B 334 


403 4 D 


i 300 


47 




B5| 39448 


9 89 4 045 


9 88 


46 








9 8 9403 


9 76 


4 










868 ,4 098 
91856 41125 


a 64 

91 52 


44 

43 


1 


1 




.V. 91845 4llSi 


91 40 


43 








-191S33 41178 


9! 28 


41 








lis! 91822 !|41 204 


91 IG 


40 


i 


o.on . " *> 






35 


9 8ia|;4 331 


91104 


39 




M80 33 48 364 






^ ,39661 


S 799I4 ai>7(H09a 


38 


SJ 


34830 38 36 S 


f<3 6 


aeus 


9 466 39688 


» 78714 a84j9IU«0 


J7 


U 


34Sa 9 a 3(>4SS 


93 OS 


38 


9-4Sa| 39716 


a 775 4 3lo;yi06S 


J6 


85 




3005 


38 A 


9 44*139741 


9 764 4 337|9HI56 


35 


S 




4 


38 G 


9 432, 39 08 


91752 41363 1IUM4 


J-l 






W fiS 


9S43 39 95 


91741 .413911, aiU3^ 


13 






38! 6 


9 4 O' 39 32 




JS 






38 4 


933 9 39S48 


)1718 4144.1 ilioiit* 








8368 




91706 !4Hll9!ill>HH6 








JH^K 


)336 


39 -K-n'-'.i ';:': ..'" '■'.'.. 


^ 






8340 


335 


39 :.-V: ■ . 








38 


9 43 


;;:,■!.: .'■'.' 


'::i 


' 




.. ■■'-,;. 11,1,1 '.r-.-l 


8 






i'l',ai555 


41813{Q0839 


5 




4^191543 


II R4U 90886 


6 




_J;391531 


11866 90814 


B 


i 




iisTa 


S0803 


-4 


i 


■;M 111608 


41910 


00790 


8 




-.r>'!tl496 


41945 


90778 


3 




"I'd 1484 


11973 


U07SS 


1 






11908 


90753 







::1 nM61 


13024 


UU741 


9 












» 


& 


3 


- :'!!■:, y:ir.-imjn 


7 


M 


SflH 4 9 






bb 


3fi 


B 




5 


fiS 


35 £8 


9 






fiT 


SS BB 


9 


■li !ii., '"' i!Ji-;i 90i;6!- 




M 


'ift 8 


93 913 


-Jli;ti:i7H,H*J109l9<lfi55 




SU 


35S 


93868 3 «^| a) lUM U 063 40617 ai366 4^336 906*3 




M 


N B NSllN B fTa [In a |kh ||N(aT| w.h. 


N.fB.l N.8. 


a 


«9 Dan V en Deir H bT Use i BH Doe- 


-jT 


>eg- 



70 



A TABLE OF NATURAL SINKS. 



M 



1 

2 

3 

4 

6 

6 

7 

8 

9 

10 

11 

12 

13 

14 

16 

16 
17 
18 
19 
20 
21 
22 
23 
24 
25 
26 
27 
28 
29 
80 

31 

32 

33 

34 

35 

36 

37 

38 

39 

40 

41 

42 

43 

44 

45 

46 
47 
48 
49 
50 
51 
52 
53 
54 



25 De^. 



N. 8. N.C8. 

90631 
90618 
90606 
90594 
90582 
90569 
90557 
90545 
90532 
90520 
90507 
90495 
90483 
90470 
90458 
90446 



42262 
42288 
42315 
42341 
42367 
42394 
42420 
42446 
42473 
42499 
42525 
42552 
42578 
42604 
42631 
42657 



42683 
42709 
42736 
42762 
42788 
42815 
42841 
42867 
42894 
42920 
42946 
42972 
42999 
43025 
43051 



43077 
43104 
43130 
43156 
43182 
43209 
43235 
43261 
43287 
43313 
43340 
43366 
43392 
43418 
43445 



90433 
90421 
90108 
90396 
90383 
90371 
90358 
90346 
90334 
90321 
90309 
90296 
90284 
90271 
90259 



43471 
43497 
43523 
43549 
43575 
43602 
43628 
43654 
43680 



56143706 



56 
57 
68 
59 

M 



43733 
43759 
43785 
43811 



90246 
90233 
90221 
90308 
90196 
90183 
90171 
90158 
90146 
90133 
90120 
90108 
90095 
90082 
90070 

90057 
90045 
90032 
90019 
90007 
89994 
89981 
89968 
89956 
89943 
89930 
89918 
89905 
89892 



26 Deg, 



N. 8. N.C8 . 

89879 
89867 
89854 
89841 
89828 
89816 
89803 
89790 
89777 
89764 
89752 
89739 
89726 
89713 
89700 
89687 



43837 
43863 
43889 
43916 
43942 
43968 
43994 
44020 
44046 
44072 
44098 
44124 
44151 
44177 
44203 
44229 



44255 
44281 
44307 
44333 
44359 
44385 
44411 
44437 
44464 
44490 
44516 
44542 
44568 
44594 
44620 

44646 
44672 
44698 
44724 
44750 
14776 
44802 
44828 
44854 
44880 
44906 
44932 
44958 
44984 
45010 



N.CS.i N.S. 



64 Deg, 



4503*^ 
45062 
45088 
45114 
45140 
45166 
45192 
45218 
45243 
45269 
45295 
45321 
45347 
45373 



89674 
89662 
89649 
89636 
89623 
89610 
89597 
39584 
89571 
89558 
0^9545 
89532 
89519 
89506 
89^493 

S9480I 

89467 

89454 

89441 

89428 

89415 

89402 

89389 

89376 

89363 

893o0 

89337 

89324 

8^311 

89298 

89285 



27 Deg, 



N. S. N. C6. 



45399 
45425 
45451 
45477 
45503 
45529 
45554 
45580 
45606 
45632 
45658 
45684 
45710 
45736 
45762 
45787 



45813 
45839 
45865 
45891 
45917 
45942 
45968 
45994 
46020 
46046 
46072 
46097 
46123 
16149 
46175 



46201 
46226 
46252 
46278 
46304 
46330 
46355 
4G381 
46407 
46433 
46458 
46484 
46510 
46536 
46561 



89101 
89087 
89074 
89061 
89048 
89035 
89021 
89008 
88995 
88981 
88968 
88955 
88942 
88928 
88915 
88902 



88888 
88875 
88862 
88848 
85835 
88822 
88808 
88795 
88782 
88768 
88755 
88741 
88728 
88715 
88701 



88688 
88674 
88661 
88647 
88634 
88620 
88607 
88593 
88580 
88566 
88553 
88539 
88526 
88512 
88499 



46587 
89272;!46613 



N.C8. 



89259! 
89245! 
89232 j 
892191 
892061 
89193 
89180 
89167 
89153 
89140 
89127 
89114 



N.8. 



63 Deg. 



46039 
46664 
46690 
46716 
46742 
46767 
46793 
4G819 
46844 
46870 
46896 
4692 1 
N.C8. 



^8 Deg. 



N. 8. N. C8. 



46947 
46973 
46999 
47024 
47050 
47076 
47101 
47127 
47153 
47178 
47204 
47229 
47255 
47281 
47306 
47332 



47358 
47383 

47409 
47434 
47460 
47486 
47511 
47537 
47562 
47588 
47614 
47639 
47665 
47690 
47716 



88485 
88472 
88458 
88445 
88431 
88417 
88404 
88390 
88377 
88363 
88349 
88336 
88322 
88308 



47741 
47767 
47793 

47818 
47844 
47869 
47895 
47920 
4794 G 
47971 
47997 
4S022 
48048 
48073 
48099 



N.8. 



62 Deg. 



48124 

48150 

48175 

48201 

43226 

48252 

48277 

48303 

48328 

48354 

48379 

4S405 

48430 

4845^ 



88295 

88231 

88267 

88254 

88240 

88226 

88213: 

88199 

88 185 

88172 

88158 

88144 

88130 

88117 

88103 

88089 



88075 
88062 
88048 
88034 
88020 
88006 
87993 
87979 
87965 
87951 
87937 
87923 
87909 
87896 
87882 



87868 
87854 
87840 
87826 
87812 
87798 
87784 
87770 
87756 
S7743 
87729 
87715 
87701 
876S7 
87673 



87659 
S7645 
87631 
87617 
87603 
87589 
87575 
87561 
',87546 
87532 
87518 
37504 
87490 
87476 



29 Deg. 



N.8. N.C8. 



48481 
48506 
48532 
48557 
48583 
48608 
48634 
48659 
48684 
48710 
48735 
48761 
48786 
48811 
48837 
48862 



48888 
48913 
48938 
48964 
48989 
49014 
49040 
49065 
49090 
49116 
49141 
49166 
49192 
49217 
49242 



49268 
49293 
19318 
19344 
49369 
49394 
49419 
49445 
49470 
49495 
49521 
49546 
49571 
49596 
49622 



N. OS. N. S. 
I 61 Deft. 



87462 
87448 
87434 
87429 
87406 
87391 
87377 
87363 
87349 
87335 
87321 
87306 
87292 
87278 
87264 
87250 



49647 
49672 
49697 
49723 
49748 
49773 
49798 
49824 
49849 
49874 
49899 
49924 
499.50 
49975 



87235 
87221 
87207 
87193 
87178 
87164 
87150 
87136 
87121 
37107 
87093 
87079 
87064 
87050 
87036 

87021 
87007 
86993 
86978 
86964 
86949 
86935 
86921 
86906 
86892 
86878 
86863 
86849 
86834 
86820 

86805 
86791 
86777 
86762 
86748 
86733 
86719 
86704 
86690 
86675 
86661 
86646 
86632 
86617 



N.CS. N.S. 



M 

60 
59 
68 
67 
66 
65 
64 
63 
62 
61 
60 
49 
48 
47 
46 
46 

44 
43 
42 
41 
40 
39 
38 
37 
36 
36 
34 
33 
32 
3i 
30 

29 
28 
27 
26 
25 
24 
23 
22 
21 
20 
19 
18 
17 
16 
15 

14 

13 

12 

11 

10 

9 

8 

7 

6 

6 

4 

3 

2 

_1 

M 



60 Deg. 



A lAMLB or KATD&AL S 



I 30 U eg 11 31 Ueg li 32 Ue^ |l 3J P eg Ij 34 Peg I 

KOO Ui l3U4|'i I J 'flT'i W 4 '■ ■■ JUl|flO 







S 3h 


TifiS 






iK loisis'ie 


^S'SW 




iH>' J 


7 '^21 fieSJitI 


o36) 



A TABLE OP MATV8AL SDIU. 



u 


a6bcB 


—sT 


lies \ •i^ ii^e \ MU«e \ 39 ueg 


2 




J1^|5L51 


IT? 


\ Jj N"8rrN(.S NS |N LS N 8 |N CS. 




t 


ftTasaaiats 


6W9 


80JI leOH'>7J4hll.,l')R( 7 lUlle 11 '777LJ60 






6738 1 


81899 


58S0 


- ish ( 1 -.1- s -^ 1 1! h t b1659 




s 


S740' 


tlMS. 


58 -4 


1 1 (r,58 




3 




81865 


5S 1 


)lw 




4 




8184d 


i8h 




^8 




& 




8183- 


o.'J« 




S5 




G 




8181 


oH) 








7 

! 
li 


^^5U 

'iTMd 

67696 

57619 


81798 
8178- 
SI765 


5^ 
5901 


,i 


53 




IS 


5764T 






48 




13 


3766 1 




IT 




14 


67891' 




48 




IS 


6T71 




15 




16 






U 




IT 


57T62S1IHI||'JH1 








IS 


67786 aiGUiSs 1 








19 


57810 8 597 59 








30 


57833 8 5a0 59 4 








SI 


57857 S 56r 1 




J9 




SS 


'i7Sa|l'' M« -^ 








23 
34 


57B01 1 


'; 


37 
38 




S3 




3a 




SB 




(31 




ST 




|-)33 




S8 


bBOil 1 




■la 




SB 


5801 




31 




80 


58070 




50 














31 


68091 




29 




33 


6811S 




"5 




83 


6814]! 




r 




34 


5B16 








3S 










36 






4 




37, 1, 


jl 








Bll 




























44 59101 








4oSt4 








46 5S44 1 
47jtl ' 


„1S 

1 . 1 ^ Alt 






3 Jl ilill J 310 








Ot,|77!1»7 10 6 7079 








aS7(S 1 407'* fiTTS 








ll/TIRll 1100 76754 








1 n bll33 b736 








1 1115 7P717 
















,64190 n670 








'nl,l" 6661 








1 1 4231|(664" 






Oh^ ,W|011|rill 


|J8H1(I n|8«ll( (JO 7Tl|ni2'i6|Tlj6''3 






SI N"^^'Xir |nXs 


l-N-e^llN-^l-NV STil Nl" ncsItTS- 


M 




~64DBff ll~fir 


D«g i S2Dep 51 Dec 60 Dee 







A Table or rATURAZ anes 





^ 


-«isr- 


, 4 D e f 4« IMg 1 4J U c 


4Ue 


"I 






N a {Ncs 


N8. N SJ NB |XC8|- B 1 CB 


S M 






*0 


b^l 6 


]6 








3 
4 
5 
6 


r 










e 

9 


M 












4 
5 


6- 




6 3 6 
69 G8 G 
9 _GJ) 


18 
47 

6 
45 






S 






9 


[4 






3 

B 
6 

8 

3 

5 


4 1 
1 


1 6 S si »6 
1 1 file 30' 6 


9 

-^ : 

B 

e 


39 
38 

6 
3S 
34 
33 
3S 
3 












1 3 3 


38 
SG 






S 


15 


r 




83 

a 


9 






; 


^ 




i 


" 






1 

6 


— 




9t 


6 
3 






BO 


1 •! 


1 
1 ^ 1 




a 






6 

5f 


" 


' 


i 
34 


e 






G 

a 

u 


1 

N e 1 N B 


llEH 3 43 i HB200, 


3 3 

2 S 








46D-K. 


'1 4R Dai;. 47 Dor t. 


lii_ 


, 





TRAVERSE TABLE 



TO EVERY QUARTER POINT OF THE COMPASS. 



TRAVBR0B TABLE. 



Oif. qflal. 4- dtp. for ( Point. 


&y. qffai. 4- dq. >7- 1 Point ( T. IV. 




Uml. 


Ul. 


iep. 


tilt. 


llL 


■*■»■- 


iMl. 


lat. 


i.,.. 


di.1. UL 


ixp. 






01.0 


00.0 


61 


60.9 


03.0 


1 


01.0 


00.1 


61 


60.7 


06.0 




3 


02.0 


00.1 


62 


61.9 


03,0 


2 


02.0 


00,2 


6) 


61.7 


06.1 




3 


03.0 


00.1 


63 


62 9 


03.1 


3 


03.0 


00.3 


63 


62.7 


062 




4 


04.0 


00.2 


64 


63 9 


03, 




04.0 


00.4 


64 


63.7 


06.3 




e 


06.0 


00.2 


66 


64,9 


03.2 


S 


05,0 


00,5 


(& 


64.7 


064 




s 


06.0 


00.3 


66 


66.9 


05.2 


6 


06.0 


006 


6£ 


65.7 


06.5 




7 


07.0 


00.3 


67 


66.9 


03.3 




07.0 


00.7 


6f 


66.7 


06.6 




e 


OBO 


00.4 


68 


67.9 


03.3 


S 


08.0 


ooa 


ed 


67.7 


06.7 




9 


09.0 


00.4 


69 


68.9 


03.4 


9 


09.0 


00.9 


69 


68.7 


06.8 




10 


10.0 


00.6 


70 


69.9 




10 


10.0 


DI.O 


71) 


69.1 


06^ 




II 


11.0 


0D.6 


71 


70.9 


03.5 


11 


10.9 


01, i 


71 


70.7 


07:0 




12 


12.0 


00-6 


72 


71.9 






11.9 


01.2 




71.7 


07.1 




13 


13.0 


00.6 


73 


72.9 


03.6 


13 


12.9 


01,3 


73 


72.7 


07.2 




14 


14.0 


00.7 


74 


73.9 


03.6 


14 


139 


01.4 


Ik 


73,6 


07.3 




16 


16.0 


00.7 


76 


74,9 


03.7 


15 


14.9 


01.5 


7S 


74.6 


07.4 




16 


16.0 


ooa 


76 


76 8 


03.7 


16 


16.9 


01.6 


7fi 


75.6 


07.4 




IT 


17.0 


OO.B 


77 


76.9 


03.8 


17 


16.9 


01.7 




76.6 


076 




18 


18 


00.9 


78 


779 


03.8 


IS 


17.9 


Ol.B 




77.6 


07.6 




19 


19.0 


00,9 


79 


TB.9 


03,9 


19 


18.9 


01.9 


73 


78.6 


07.7 




•a 


20.0 


01.0 


80 


79.9 


03,9 


20 


19.5 


02.0 


80 


79.6 


07,8 




21 


21.0 


01.0 


Bl 


80.9 


04.0 


21 


20.9 


02,1 


81 


S0.6 


07.9 




M 


22.0 


01.1 


B2 


ai.9 


04,0 


22 


21.9 


02.2 


Bl 


81.6 


oe.0 




23 


23.0 


01.1 


83 


82.9 


04 


23 


22-9 


02.3 


BJ 


82.6 


08.1 




24 


24.0 


01.2 


S4 




W.l 




23.9 


02.4 


S4 


83.S 


0&2 




26 


2S.0 


01.2 


86 


wig 


04.2 


25 24.9 


02,4 




84.6 


08.3 




26 


26.0 


01.3 


86 


85.9 


04,2 


26 26,9 


02.5 




65.6 


oa4 




27 


27.0 


01.3 


87 




04.3 


27 26 9 


02.6 


87 


S6.6 


086 




28 


2ao 


014 


88 


87!9 


04,3 




02,7 


83 


87.6 


oe.6 




29 


29.0 


01.4 


89 


88.9 


04.4 




02.8 


39 


88.6 


08.7 




30 


30,0 


01.6 


90 


89.9 


04.4 


30 29.9 


02,9 


90 


89.6 


oa8 




31 


31.0 


01.6 


91 


909 


04.6 


31 30,9 


03.0 


91 


90.6 


03.9 




32 


32.0 


01.6 


92 


91.9 


04.6 


32 31.8 


03.1 


92 


916 


09.0 






33.0 


01,6 


93 


92.9 


04.6 


33 32,8 


03.2 


93 


92.6 


09.1 




34 


34.0 


01.7 


94 


93.9 


04.6 


34 33.8 


03.3 


94 


93.6 


09.2 




3S 


36.0 


01.7 


96 


94.9 


04.7 


■& 34.8 


03.4 


95 


94.5 


09.3 




36 


36-0 


01.8 


96 


95.9 


04.7 


36 36.B 


03.5 


96 


95.6 


09.4 




37 


37.0 


01,8 


97 


96.9 


04.B 


37 36.8 


03.6 


97 


966 


09.6 




38 


38.0 


01.9 


98 


97,9 


04.8 


38 37.8 


03.7 


9B 


97.6 


09.6 






39.0 


01.9 


99 


98.9 


04.9 


39^3a8 


03,8 




98.6 


09.7 




40 


40.0 


02.0 


100 


99.9 


04,9 


40 39.8 


039 


100 


99.5 


09.8 




41 


4i.O 


O'J.O 


01 


100.9 


05.0 


41 40.8 


04.0 


lOt 


10O.6 


099 




42 


41.9 


Ki.l 


102 


lOI.S 


05.0 


12 41.8 


04.1 


m 


101.6 


10.0 




43 


42 9 


Oil 


103 


102.9 


05.1 


43 42.8 


04,2 


103 


102.6 


101 






43.9 


02.2 


04 


103.9 


05.1 


44 43.8 


04.3 


104 


103.6 


10.2 




46 


44.9 


B2.2 


105 


104.9 


05.2 


45 44.8 


04,4 


105 


104.B 


10.3 




46 


46.9 


02.3 


06 


106.9 


05.2 


46 , 45.8 


04.5 


106 


106.5 


10.4 




47 


46.9 


02.3 




106.9 


03.3 


47 ; 46.8 


04.6 




106.6 


10.6 




48 


47.9 


02.4 


108 


107.9 


05.3 


4B 47.8 


04.7 


108 


107.6 


10.6 




49 


43.9 


02.4 


109 


108.9 


05.4 


49 4a8 


04.8 


109 


108.6 


10,7 




50 


49.9 


02.6 


110 




05.4 


SO 49.8 


049 


110 


109.6 


10,8 




51 


60.9 


02.6 


111 


110.9 


0E.5 


5! 50.8 


05.0 


111 


1106 


0.9 




B2 


B1.9 


02.6 


112 


111.9 


0S.5 


52 ■ 61.7 


06.1 


iia 


111.6 


1.0 




S3 


62.9 


02.6 


113 


112.9 


05.S 


53 62.7 


05.2 


113 


112.6 






64 


B3.9 


02.7 


114 


113,9 


066 


54 63.7 


05.3 


114 


113.5 






es 


64.9 


02,7 


115 


114.9 


05.6 


55 B4.7 


05.4 


115 


tl4.5 


1L3 




66 


65,9 


02 8 


116 


115.9 


05.7 


5(i 55.7 


05.B 


116 


1 16.4 


11.4 




67 


56.9 


028 


117 


116,9 


05.7 


57 66.7 


05,6 


117 


116.4 


11.6 




68 


67.9 


029 


U8 


117.9 


05.8 


53 67.7 


05.7 


118 


117.4 


11.6 




6S 


68.9 


02.9 


119 


118.9 


05.8 


69 6a7 


05.8 


119 


iia4 


11.? 




60 


69-9 


02.9 


120 


119.9 


06.9 


60 ; 69.7 


05.9 


120 


119.4 


11.8 




IM. 


iV 




iiM. 


^■ 




4i»L\ i^ 






for 71 P«n«. » for 7* Point*. 





TRAVEBSE TABLE. 



77 



T. iv.J Dif. <^laL ^ dep.foriPoinU, 1 


Dif:qfkU.4'depJoriJPoinL | 


igt. 


UL 


Ay. 


di9L 


ImL 


Ar. 


duL 


fat 


Asp. 


diaL 


tat. 


dtp. 


1 


«1.0 


00.1 


61 


6a3 


06.9 


1 


01.0 


00.2 


61 


59.6 


11.9 


2 


02.0 


00.3 


62 


61.3 


09.1 


2 


02.0 


00.4 


62 


60.8 


18.1 


3 


03.0 


00.4 


63 


62.3 


09.2 


3 


02.9 


00.6 


63 


61.8 


12.3 


4 


04.0 


00.6 


64 


63.3 


09.4 


4 


03.9 


00.6 


64 


62.8 


12.6 


5 


049 


007 


65 


64.3 


09.5 


5 


04.9 


01.0 


66 


63.7 


12.7 


6 


05.9 


00.9 


66 


65.3 


09.7 


6 


05.9 


01.2 


66 


64.7 


119 


7 


06.9 


01.0 


67 


66.3 


09.8 


7 


06.9 


01.4 


67 


66.7 


13.1 


8 


07.9 


01.2 


68 


66.3 


10.0 


8 


07.8 


01.6 


66 


66.7 


13.3 


9 


oao 


01.3 


69 


68.2 


10.1 


9 


06.8 


01.6 


69 


67.7 


13.6 


10 


09.9 


01.5 


70 


69.2 


10 3 


iO 


09.8 


02.0 


70 


66.7 


13.7 


H 


10.9 


01.6 


71 


70.2 


10.4 


11 


10.8 


02.1 


71 


69.6 


13.9 


12 


11.9 


01.8 


72 


71.2 


10.6 


12 


11.8 


02.3 


72 


70.6 


14.0 


13 


12.9 


01.9 


ra 


72 2 


10.7 


13 


12.7 


02.5 


73 


71.6 


14.2 


14 


13.8 


0^1 


74 


73.2 


10.9 


14 


13.7 


02.7 


74 


72.6 


14.4 


15 


14.8 


02.2 


75 


74.2 


11.0 


15 


14.7 


02.9 


75 


73.6 


14.6 


16 


15.8 


02.3 


76 


75.2 


11.1 


16 


15.7 


03.1 


76 


74.6 


14.8 


17 


16.8 


02.5 


77 


76.2 


11.3 


17 


16.7 


03.3 


n 


76.6 


15.0 


18 


17.8 


02.6 


78 


77.2 


11.4 


18 


17.7 


03.5 


76 


76.5 


15.2 


19 


ia8 


02.8 


79 


78.1 


11.6 


19 


18.6 


03.7 


79 


77.5 


16.4 


20 


19.8 


02.9 


80 


79.1 


11.7 


20 


19.6 


03.9 


80 


78.6 


16.^ 


21 


20.8 


03.1 


81 


80.1 


11.9 


21 


20.6 


04.1 


61 


79.4 


16.6 


22 


21.8 


03.2 


82 


81.1 


12.0 


22 


21.6 


04.3 


62 


80.4 


16.6 


23 


22.7 


03.4 


83 


82.1 


12.2 


23 


22.6 


04.5 


63 


61.4 


16 2 


24 


237 


.03.5 


84 


83.1 


12.3 


24 


23.5 


04.7 


64 


82.4 


16.4 


25 


24.7 


03.7 


85 


84.1 


12.5 


25 


24.5 


04.9 


65 


63.4 


16.6 


26 


25.7 


03.8 


86 


85.1 


12.6 


26 


25.5 


05.1 


66 


84.3 


16.6 


27 


26.7 


04.0 


87 


86.1 


12.8 


27 


26.5 


05.3 


67 


85.3 


17.0 


28 


27.7 


04.1 


88 


87.0 


12.9 


28 


27.5 


05.5 


66 


86.3 


17.2 


29 


28.7 


04.3 


89 


88.0 


13.1 


29 


28.4 


05.7 


69 


67.3 


17.4 


30 


297 


04.4 


90 


89.0 


13.2 


30 


29.4 


05.9 


90 


86.3 


17.6 


31 


30.7 


04.5 


91 


90.0 


13.3 


31 


30.4 


06.0 


91 


69.2 


17.6 


32 


31.7 


04.7 


92 


91.0 


13.5 


32 


31.4 


06.2 


92 


90.2 


16.0 


33 


32.6 


04.8 


93 


92.0 


13.6 


33 


32.4 


06.4 


93 


91.2 


16.1 


34 


33.6 


05.0 


94 


93.0 


13.8 


34 


33.3 


06.6 


94 


92.2 


16.3 


35 


34.6 


05.1 


95 


94 


13 9 


35 


34.3 


06.6 


95 


93.2 


ia6 


36 


'35.6 


05.3 


96 


95.0 


14.1 


36 


35.3 


07.0 


96 


94.2 


la? 


37 


36.6 


05.4 


97 


95.9 


142 


37 


36.3. 


07.2 


97 


95.1 


lao 


38 


37 6 


05.6 


98 


96.9 


14.4 


38 


37 3 


07.4 


96 


96.1 


19.1 


39 


38.6 


05.7 


99 


97.9 


14.5 


39 


38.2 


07.6 


99 


97.1 


19.3 


40 


39.6 


05.9 


100 


9a9 


14.7 


40 


39.2 


97.6 


100 


9ai 


19.B 


41 


40.6 


06.0 


101 


99.9 


14.8 


41 


40.2 


08.0 


101 


99.1 


19.7 


42 


41.5 


06.2 


102 


100.9 


15.0 


42 


41.2 


oa2 


102 


100.0 


19.9 


43 


42.5 


06.3 


103 


101.9 


15.1 


43 


42.2 


08.4 


103 


101.0 


20.1 


44 


43.5 


065 


104 


102.9 


16.3 


44 


43.2 


06.6 


104 


102.0 


20.3 


45 


44.5 


06.6 


105 


103.9 


15.4 


45 


44.1 


06.6 


106 


103.0 


20.6 


46 


45.5 


06.7 


106 


104.8 


15.5 


46 


45.1 


09.0 


106 


104.0 


20.7 


47 


46.5 


06.9 


107 


103.8 


15.7 


47 


46.1 


09.2 


107 


104.9 


20.9 


48 


47.5 


07.0- 


108 


106.8 


15.8 


48 


47.1 


09.4 


106 


105.9 


21-1 


49 


48.5 


07.2 


109 


107.8 


16.0 


49 


48.1 


09.6 


109 


106.9 


21.3 


50 


49.5 


07.3 


110 


108.8 


16.1 


60 


49.0 


09.6 


IM) 


107.9 


21.6 


51 


50.4 


07.5 


111 


109.8 


16.3 


51 


50.0 


10.0 


111 


106.9 


21.7 


52 


51.4 


07.6 


112 


110.8 


16.4 


52 


51.0 


10.1 


112 


109.8 


21.9 


53 


52.4 


07.8 


113 


111.8 


1&6 


53 


62.0 


10.3 


113 


110.8 


22.0 


54 


53.4 


07.9 


114 


112.8 


16.7 


54 


53.0 


10.5 


114 


111.8 


22.2 


55 


54.4 


08.1 


115 


113.7 


16.9 


55 


53.9 


10.7 


115 


112.8 


22.4 


56 


55.4 


08.2 


116 


1147 


17.0 


56 


54.9 


10.9 


116 


113.8 


22.6 


57 


56.4 


084 


117 


115.7 


17.2 


57 


S5.9 


11.1 


117 


114.7 


22.8 


58 


57.4 


06.5 


118 


116.7 


17.3 


68 


56.9 


11.3 


116 


115.7 


23.0 


59 


5a4 


08.7 


119 


117.7 


17.6 


69 


67.9 


11.6 


119 


116.7 


23.2 


60 


59.3 


06.8 


120 


118.7 


17.6 


60 


66.6 


11.7 


120 


\r 


iUL dtp. 


UlL 


iUL 


^ 


tat. 


iM. 


tew tat 1 


dtiL 




for 7^ 


Pcint 


•. 




ftNrT 


Pflfat 







TaTBXBE TABLB. 



Dif. qflat. 4- dep.fvT 1( i*«n(s. 


D}f.BflalA 


^dtf.for\\Point,.[T.n.\ 


tin. 


UL 


iv. 


iilL 


lat. 


dip. 






itf. 






irf. 


1 


DI.0 


D0.2 


61 


59.2 


4,6 


1 


01.0 


00.3 


61 


68,4 


17.7 


2 


OI.S 


D0.5 


62 


60,1 


5.1 


2 


01,9 


00.6 


62 


59.3 


19 




02.9 


0O.J 


63 


61.1 


S.3 




OZ-9 


D0.9 


63 


eo.3 


las 




0J.9 


01.0 


64 


62.1 


5.6 


4 


03.8 


01.2 


64 


61.2 


18-6 


B 


04.9 


01.2 


65 


63,1 


5.8 


6 


04,8 


01.6 


66 


62.2 


ia9 


6 


05.8 


01.5 


«e 


«4,D 


6.0 


6 


05.7 


01.7 


66 


63.2 


19.2 


7 


oa.s 


01.7 


67 


65.0 


6.3 


7 


06.7 


02.0 


67 


64.1 


19.4 


e 


07.9 


01.9 


68 


66 


e.s 


e 


07.7 


02,3 


69 


66.1 


19.7 


9 


05.7 


02,2 




66.9 


6.8 


9 


oe.6 


02.6 


69 


66.0 


20 


10 


09.7 


02,4 


TO 


67.9 


J.O 


10 


09.6 


02.9 


70 


67.0 


20.3 


I) 


10.7 


0J,7 


71 


68.9 


17.3 


11 


10.5 


03.2 


71 


67.9 


20.6 


11 


11.6 


0-J.9 




69.8 






11.6 


03.5 


72 


68.9 


S0.9 


)3 


12.6 


03.2 


73 


70.8 




13 


12.4 


03,8 


73 


69.9 


21.2 


14 


13« 


03.4 


74 


71.8 


18.0 


14 


13.4 


04.1 


74 


70.8 


21.6 


IB 


14.6 


036 


75 


72.8 


ia-2 


16 


14.4 


04,4 


76 


71.8 


21.8 


16 


15 S 


03.9 


76 


73.7 


ia5 


16 


15.3 


04.6 


76 


72.7 


22.1 


17 


16.6 


04,1 


77 


74.7 


1S.7 


17 


16.3 


04.9 


77 


737 


22.3 


IS 


17.5 


04,4 


78 


75.7 


19-0 


18 


17 2 


05.2 


78 


74 6 


22.6 


19 


ia4 


01.8 


79 


76.6 


19,2 


19 


ia.2 


Dj.5 


79 


75.6 


22.9 


20 


19.4 


04,9 


BO 


77.6 


19,4 


20 


19.1 


05.8 


60 


76.6 


23 2 


21 


20.4 


05.1 


91 


78,6 


19.7 


21 


20.1 


06.1 


81 


77.5 


23.5 


22 


21.3 


OJ.3 


G2 


79.5 


9.9 


22 


21.1 


06.4 




78.5 


23.9 


23 


22.3 


03,6 




80.6 


0.2 


23 


22,0 


06.7 




79.4 


24.1 


24 


23.3 


05.B 


94 


81.5 


0.4 


24 


23.0 


07.0 


84 




24.4 


25 


24.3 


06.1 


86 


82.5 


0.7 


26 


23.9 


073 


85 


eL3 


24.7 


38 


25.2 


06,3 


86 


83.4 


20.9 


26 


24.9 


07.5 


86 




25.0 


27 


2.26 


06,6 


87 


S4.4 


21.1 


W 


25.8 


07.B 


87 


83!3 


25.2 


2a 


27.2 


06,9 


88 


85. 4 


21.4 


28 


26.8 


08.1 




84.2 


25.6 




28.1 


07.0 


89 


86.3 


21.6 


29 


27.8 


09.4 




86.2 


25.8 


30 


29.1 


07.3 


90 


87.3 


21.9 


30 




08.7 


90 


86.1 


26,1 


31 


30.1 


07.6 


91 


8B.3 


22.1 


31 


29.7 


09-0 


91 


87.1 


264 




31.0 


07.8 


92 


99.2 


22.4 


32 


30.6 


09.3 


92 


BS.0 


26.7 


33 


32.0 


03,0 




90,2 


22.6 


33 


31.6 


09-6 


93 


B9.0 


27.0 


34 


33.0 


09.3 


94 


91.2 


22.8 


34 


32.5 


09.9 


94 


90.0 


27.3 


35 


34.0 


08,6 


96 


9i2 


23.1 


35 


33 5 


10 2 


95 


90.9 


27.6 


37 


34.9 


03.7 


96 


93.1 


23.3 


36 


34.S 


10.4 


96 


91.9 


27.9 


37 


36,9 


09-0 


97 


94.1 


23-6 


37 


35.4 


10.7 


97 


92-8 


29.2 


3S 


36.9 


09.2 


98 


96.1 


23.B 


39 


36-4 


il.O 




93-3 


28.4 


39 


37.8 


09,5 


99 


96.0 


24.1 




37.3 


11.3 




94.7 


28.7 


40 


38.8 




lOO 


97.0 


24.3 


40 




11.6 


100 


95.7 


29.0 


41 




10,0 


101 


99.0 


24,5 


41 


39.2 


11.9 


101 


96.7 


29.3 


42 


40J 


10,2 


IM 


98.9 


21.8 


42 


40.2 


122 


102 


97.6 


29.6 


43 


41.7 


10,4 


103 


99.9 


25.0 


43 


41.2 


12.6 


103 


98.6 


29.9 


44 


42.7 


10,7 


104 


100.9 


25.3 


41 




12.9 


104 


99.5 


30.2 


45 


43.7 


10-9 


'2^ 


101.9 


25.6 


45 


43! 1 


13.1 


m, 


100.5 


30-6 


4S 


44.6 


11.2 


106 


lOi.8 


25.8 


46 


44.0 


13.3 


106 


101.4 


30.8 


47 


45.6 


11,4 


107 


103.B 


26.0 


47 


45.0 


13-6 


107 


102.4 


31,1 


48 


46.6 


11-7 


loa 


104.8 


262 


48 


45.9 


13.9 


lOa 


1034 


31.4 


49 


47.5 




109 


105.7 


2e.5 


49 


46.9 


14.2 


109 


104.3 


316 


50 


48.6 


vi-i 


110 


106.7 


26.7 


50 




14.6 


110 


105.3 


31.9 


51 


49.6 


12.4 


111 


10V.7 


27.0 


El 


43,8 


14-8 


111 


106.2 


32.2 


52 


60.4 


12,6 


112 


103.6 


27,2 


52 


49.8 


15-1 


112 


I0T.2 


32-5 


63 


61.4 


12.9 


113 


109.6 


27.5 


53 


50.7 


15-4 


113 


(B.I 


32.8 


54 


62.4 


13.1 


114 


110.6 


■27,7 


54 


61.7 


15.7 


114 


09 1 


33.1 


65 


63.4 


13.:! 


"S 


111.6 


27,9 


65 


52,6 


16-0 


115 


10.1 


33.4 


50 


64.3 


13.6 


U6 


112.5 


2a2 


66 


53.6 


6-3 


116 


U.O 


33.7 


67 


55.3 


13.9 


117 


1 3.5 


28.4 


B7 


54-5 


6.5 


117 


120 


34.0 


68 


66.3 


14,1 


ua 


1 4.5 


2a7 




56-5 


6.9 


118 


12.9 


34.2 


59 


57.2 


14,3 


119 


1 6,4 


28.9 


59 


56.5 




119 


13.9 


34.5 


EO 


E8.2 


14.6 


m 


16.4 


29.2 


60 


57.4 


I7!4 


120 


114,8 


34.8 




if- 






dtp. 




d»(. 


Ay. 


IM. 




rfq,. 




forOlPmnti. ' forOi 


Poin' 


1 



TBATBUB TABLE. 



T.iy.]Dif.qflal.^ dip. for 1| Poinfa.' 


Dir..,flci.4-d'^-for 


2P«nl>. 




din. 


uc 


^ 


■"* '-' 


it- 


dUU Ut. i^. 


iix. 


1*1. 


**■ 






00.9 


00.3 


61 


57.4 


20.5 


1 00.9 004 


61 


se,4 


23.3 




2 


01-9 


00.7 


62 


S3.4 


20,9 


2 01.8 00.3 


62 


67.3 


23.7 




3 


02.B 


01.0 


63 


69-3 


21-2 


3 te.9 01.1 


63 


58.2 


24 1 






03.8 




64 


60.3 


21.6 


4 


03.7 01-6 


64 




24.5 




5 


04.7 


0U7 


65 


61.2 


21.9 




(M.6 01.9 


6S 


6ai 


24.9 




6 


05-6 


02.0 


66 


62.1 


212 


6 


OS-5 02.3 1 


66 


61.0 


26.3 




7 


06.6 


02.4 


67 




22.6 


7 


06.5 ■ 


02.7 


67 


61.9 


2S.6 




8 


07.S 


02.7 


68 


64^0 


22-9 


8 


07.4 


03.1 




62.8 


26.0 




9 


08.5 


03.0 




65.0 


23-2 


9 


06,3 


03.4 


69 


63.8 


26.4 




10 


09.4 


03.4 


70 


65.9 


23.6 


10 


09.2 


03.8 


70 


64.7 


26.8 






10-4 


03-7 




66.8 


23.9 


11 


10.2 


04.2 


n 


66.6 


27.2 




12 


11.3 


04.0 


72 


6T.8 


24.3 


12 


11.1 


04.6 


n 


66-6 


27.6 




13 


12.2 


04.4 


73 


68.7 


24.6 


13 


12.0 


05-0 


73 


67.4 


27.9 




14 


13.2 


04.7 


74 




24.9 


14 


1Z9 


05.4 




es.4 


28,3 




15 


14.1 


05.1 


75 


70^6 


2S.3 




13.9 


05.7 




69.3 


28.7 




16 


15.1 


05.4 


76 


71.6 


25.6 


16 


14,8 


06.1 


76 


702 


29.1 




17 


16 


05.7 




72.5 


25-9 


17 


15.7 


066 


77 


71-1 


29.B 




la 


17.0 


06.1 


78 


73.4 


56.3 


18 


16.6 


06.9 


79 




29.9 




19 


17.9 


06.4 


79 


74.4 


26.6 


19 


17,6 


07.3 


79 


73.0 


30.2 




w 


I8.S 


06-7 


80 


7S.3 


26.9 


20 


18.5 


07.7 


80 


73.9 


S0.6 




21 


19.8 


07.1 


81 


76-3 


27.3 


21 


19,4 


08,0 


81 


74.8 


310 




22 


20,7 


07.4 


82 


77.2 


27.6 


22 


20.3 


0B,4 


92 


76,8 


31.4 




23 


21.7 


OT-7 




7a 1 


28.0 


23 


21,3 


oae 


83 


76.7 


31.8 




•u 


22-6 


DS.1 


84 


79,1 


28.3 


24 


22-2 


09.2 


84 


77-6 


32.1 




25 


23.5 


oe.4 


85 


E0.0 


28.6 


25 


23 1 


09.6 


85 


78.5 


32.5 




26 


24.S 


03.B 


86 


81-0 


29.0 


26 


24 


10,0 


86 


79,6 


32.9 




27 


25.4 


09.1 


87 


81.9 


29.3 


« 


24.9 


10,3 


87 


80.4 








26.4 


09.4 


88 


B2.9 


29,6 


28 


25,9 


10,7 




81-3 


33.7 




29 


27-3 


09-8 


89 


83.8 


30.0 


29 


26-8 


11.1 






34.1 




30 


28.2 


10-1 


90 


84.7 


30.3 


30 


27,7 


II.B 


90 




34.4 




31 


29.2 


10.4 


91 


85.7 


30-7 


31 


29.6 


11,9 


91 


S4.1 


34.8 




33 


30.1 


10.8 


92 


86.6 


31.0 




29,6 


12.2 


92 


85-0 


36,2 




33 


31.1 


11.1 




37,6 


31.3 




30,5 


12-6 


93 


85.9 


35,6 




34 


32.0 


11.6 


94 


ee-s 


31.7 


34 


31.4 


13,0 


94 


86,8 


36.0 




35 


33-0 


11-8 


95 




32.0 


35 


32,3 


1*4 


95 


S7.8 


36,4 




36 


33 9 


12.1 


% 


90.4 


32.3 


36 


33,3 


13,3 


96 


88.7 






37 


34.B 


12-5 


97 


91.3 


32.7 


37 


34.2 


14.2 


97 


89.6 


3/1 




■i3 


35-9 


12.8 


98 


92.3 


33.0 


38 


35.1 


14.5 


98 


90,B 


37.B 




39 


36.7 


13-1 


99 


93.2 


33.3 


39 


36.0 


14,9 


99 


91.6 


37.0 




40 


37.7 


13-5 


100 


94-2 


33,7 


40 


37.0 


16.3 


lao 


92,4 


38.3 




41 


33.6 


13.8 


01 


96.1 


34.0 


41 


37.9 


1E.7 


101 


93,3 


3a7 




42 


39.5 


14.1 


02 


96.0 


34.4 


42 


388 


16,1 


102 


94.2 


39,0 




43 


40,5 


14.5 


03 


97.0 


34.7 




39.7 


I6.S 


103 


95,2 


39,4 






41-4 


14.8 


04 


97.9 


35.0 


44 


40,6 


16.B 


104 


96.1 


39,B 




46 


42.4 


15.2 


105 


•18.0 


35-4 


45 


41.6 


17.2 


105 


97.0 


40.2 




46 


43-3 


15.6 


106 


99.8 


36.7 


46 


42-5 


17.6 


106 


97.9 


4ae 




47 


44.3 


16,8 


107 


1O0.7 


36.0 




43,4 


18.0 


107 


98,9 


41-0 




4e 


452 


16.2 


103 


101.7 


36-4 


48 


44-4 


18.4 


108 


99.8 


41.3 




49 


46-1 


16.6 


109 


1016 


36.7 


49 


45-3 


18.8 


109 


100.7 


41.7 




50 


47. 1 


16.B 


no 


103.6 


37.1 


60 


46.2 


19,1 


110 


101.0 


42.1 




51 


48,0 


172 


111 


104,6 


3T.4 


61 


47.1 


19.6 


111 


102.6 


42.S 




62 


49.e 


17.5 


112 


10.^.4 




62 


48,0 


19.9 


112 


103.5 


42.9 




63 


49,9 


17.9 


113 


103.4 


3ai 


53 


49,0 


20,3 


113 


104.4 


43-2 




54 




182 




107.3 


38.4 




49,9 


30,7 


114 


105.3 


43.8 




S6 


Bi.6 


18 5 


MB 


103.3 


38.7 


55 


50.8 


2i,0 


116 


106.3 


44.0 




E6 


52.7 


IB 9 


116 


109,2 


39.1 


56 


61,7 


21.4 


116 


107-2 


44.4 




67 




19 2 


117 


110.2 


39.4 


57 


62.T 


21,8 


117 


10H.1 




68 


54!6 


19.5 




111.1 


39.7 


58 


63.6 


22 2 


lie 


109.0 


4b!! 


59 


55.5 


19.0 


119 


112.0 


40.1 


69 


54.6 


22.6 


119 


109.9 


4».l 


60 


56.5 


20.2 


120 


113.0 


40.4 


60 


65.4 


23,0 


120 


110^ 






dtp. 


bu. 




dip. 






<(«. 


III. 


an. 




IbretPoin 




rot S Poii 









BO 



TRAVBRBB TABLE. 



r. i».J Dif. i^hL4- <iipJfer2tPoin[t 



B2.4 24-B 
E3.3 25.2 
E4.2 2L7 



Di/: flThK. ^ iJijM'»r3 iPofeO. [ T-nr 



116 ; lD4,a , 49.2 

116 ' 1M.S 49,6 

117 105.3 E0.0 



1 


00.9 


49. 

0O.B 


6L 


53.8 




2 


01.8 


0U.0 


61! 


64.7 




3 


02.6 


01.0 


63 


66.6 




•1 


016 


Ol.t 


61 


fi6.4 






04.4 


02.4 


6G 


67.3 




6 


0S.3 


02.S 


66 


6a2 




7 


06.2 


03.a 


67 


59.1 






07.1 


03.a 


68 


60.0 




9 


07.9 


04.x 


69 


eas 




10 


oe-e 


04.7 


7(1 


61.7 




I! 


09.T 


06.:i 




6Z6 




12 


10;6 


oe.7 


72 


63.6 




13 


1.S 


06.1 


73 


64.4 




14 


!.3 


06.« 




65.3 




15 


12 


07.1 


75 


66.1 




16 


1.1 


07.6 


76 


67.0 




17 


16.0 


O3.0 


77 


67.9 




IB 


1B.9 


0&6 


7B 


68.8 




19 


16.8 


09.0 


7B 


69.7 




20 


17.6 


09.4 


80 


70.6 




21 


19.6 


09.9 


81 


71.4 




22 


19.4 


ia4 


82 


T2.3 




23 


20.3 


1 .8 


83 


73.2 




24 


21.2 


1 ,3 


B4 






23 


22.2 


1 .8 


85 


7B.0 


4 


26 


22.9 


1 .3 


86 


75.9 


■ 


2- 


23.8 


12.7 


87 


76.7 




28 


24.7 


13.2 


88 


77.6 


■ 


29 


26.6 


13.T 


89 


78.6 


■ 


■iO 


26.6 


14.1 


90 


79.4 




31 


27.3 


14.6 


91 


80.3 


■ 




2a2 


16.1 


92 


Bl.I 




ii 


29.1 


15.6 


93 


B2.0 


■ 




30,0 


16.0 


94 


82.9 


4 




30.9 16.B 


96 


B3.B 


4 


36 


31.8 


17.0 


96 


B4.7 


4 


17 


32.6 


17.4 


97 


86.6 


■ 




33.6 


17.9 


98 


86.4 






34.4 


18.4 


99 


87,3 


■ 


40 


36.3 


18.9 


100 


83.2 


4 




36.2 


19.3 


101 


89.1 


4 


12 


37.0 


19.8 


102 


90.0 


■ 


43 


37.9 


20.3 


103 


90.8 






38.B 


20.7 


104 


91.7 


' 


45 


39.7 


21.2 


106 


916 


4 


46 1 40.6 


21.7 


106 


93.6 


& 


47 ' 41.6 


22.2 


107 


94.4 


5 


43 1 42.3 


22.6 


108 


96.3 


6 


49 ' 43.2 


23.1 


109 


96.1 


5 


50 1 44.1 


23.6 


110 


97.0 


5 


51 1 46.0 


24.0 


111 


97.9 


6 


■,■> ' 45.9 


24.5 


112 


oas 


6 


53 1 46.7 
oJ 47.6 


25.0 
26-B 


113 
114 


99.7 
100.5 


6 

6, 


55 1 4B.5 


25.9 


]1B 


101.4 


B. 


56 49.4 


26.4 


116 


102.3 


& 


67 ' 60.3 


26.9 


117 


103.2 


51 


53 , 61.2 


27.3 


118 


104.1 


6 


69 1 62.0 


27.8 


119 


105.0 


6< 


50 , 62.9 


2a3 


120 


106.8 


ei 


i.L' <t9. 


M. 


iit. 


i.,. 




t)t6tr<mt 





XRVEftBE TABLS. 



7-.,. 


] Dif. of lot. <C d^- for 21 Point.. 


Dif. oflal.4- dtp- for 3 Point.. | 




UL 


dtp. 


iiit. 




tfO.. 


iitl. 


Jul 


rfq.. 


iui. 


Ut. 


dtp. 




00.0 


D0.5 


61 


52.3 


31.4 


1 


00,8 


00.6 


*1 


60.7 


33.9 


2 


01.7 


01.0 


62 


63.2 


31.9 


2 


01.7 


00.1 


62 


S1.S 


34.4 


3 


02.6 


01.5 


63 


54.0 


32,4 


3 


02,6 


01.7 


63 


52.4 


35.0 


4 


03.4 


02.1 


64 


54.9 


32.9 


4 


03.3 


02,2 


64 


53.2 


35.6 


B 


04.3 


02.6 


65 


65.6 


33.4 


6 


04.2 


02.8 


66 


54.0 


36.1 


6 


05.1 


03.1 


66 


66.6 


33.9 


6 


05.0 


03.3 


66 


54.9 


36.7 


7 


06.0 


03.6 


67 


67.5 


34.4 


7 


06.8 


03.3 


67 


66.7 


37.2 


B 


06,9 


04.1 




B9.3 


35.0 


8 


06.7 


04.4 


6S 


66,6 


37.9 


9 


07.7 


04.6 


69 


69.2 


35.5 


9 


07.5 


05.0 


69 


57.4 


3a3 


10 


oa.6 


05.1 


70 


60.0 


36.0 


10 


oa.3 


05,6 


70 




38.9 


1 


09.4 


05.7 


71 


60.9 


36.6 


11 


09.1 


06.1 


Jl 


59-0 




2 


10.3 


06.2 


72 


61.8 


37.0 


i2 


10.0 


06.7 


72 


59.9 


40:0 


3 


11.2 


06.7 


73 


62.6 


37.B 


3 


10.8 


07.2 


73 


60.7 


40.6 


4 


12.0 


07-2 


74 


63.5 


38.0 


14 


11.6 


07.B 


74 


61.6 


41.1 


5 


12.9 


07.7 


75 


64.3 


38.6 


16 


12.6 


oa3 


76 


62.4 


41.7 


G 


13.7 


03.2 


76 


65.2 


39.1 


16 


13.3 


0ft9 


76 


63.2 


42.2 


7 


14.6 


l».7 


77 


66.0 




17 


14.1 


09.4 


77 


64.0 


42.8 


8 


15.4 


09.3 


78 


66.9 


40[l 


19 


16.0 


10.0 


78 


64.a 


43.3 


9 


16.3 


09.fl 


79 


67.8 


40.6 


19 


15.8 


10.6 


79 


66.1 


43.9 


20 


17.2 


10.3 


BO 


69.7 


41.1 


20 


16.6 


11.1 


SO 


66.6 


44.4 


21 


lao 


10-9 


81 


69.5 


41.6 


21 


17.6 


11.7 


81 


67.3 


45.0 


22 


18.9 


11.3 


82 


70.3 


42.2 


!S 


ia3 


12,2 






45.6 


23 


19.7 


11.8 




71.2 


42.7 


23 


19.1 


12.B 






46-1 


24 


20.6 


1Z3 


84 


73,0 43.2 


24 


20,0 


13.3 


84 




46.7 


25 


21.4 


12.9 




72.9 


43.7 


26 


20.8 


13.9 


S6 


70:7 


47.2 


2fi 


22.3 


13.4 


66 


73.0 


44.2 


26 


21.6 


14.4 


ee 


71.5 


47.B 


27 


23.2 


13.9 


87 


74.6 


44.7 




22.4 


5.0 


87 


72.3 


49.3 




24.0 


14.4 




75.5 


45.2 




23.3 


6.6 


89 


73.2 


48-S 


29 


24.9 


14.9 




76.3 


46.7 




24.1 


6.1 


89 


74.0 


49.4 


30 


26.7 


15.4 


90 


77.2 


46.3 


30 


24,9 


6.7 


90 


74.9 


60.0 


31 


26,6 


15.9 


91 


79.1 


46.8 


31 


25.8 


17.2 


§1 


76.7 


60.6 


32 


27.4 


16.4 


92 


79.9 


47.3 




26.6 




92 


76.5 


61-1 


33 


28.3 


17.0 




79.S 


47.8 




27.4 


19.3 


93 


77.3 


B1.7 


34 


29.2 


17.5 


94 


80,6 


48.3 


34 


28.3 


18,9 




73.2 


62.2 


36 


30.0 


18.0 


95 


ai.5 


48.8 




29.1 


19.4 




79.0 


B2.S 


36 


30.9 


18.5 


96 


32.3 


49.3 


36 


29.9 


20.0 




79.8 


53.3 


37 


31.7 


19.0 


97 


B3.2 


49.9 


37 


30.3 


20.6 


97 


80.6 


63-9 


3a 


32.6 


19.5 


9S 


64. 1 


60.4 




31.6 


21.1 


99 


5t.B 


54.4 


39 


33-5 


20.0 


99 


64.9 


50.9 


39 


32.4 


21.7 


59 


813 


BB-0 


40 


34.3 


20.6 


100 


B5.e 


61.5 


40 


33.3 


22.2 


100 


83.1 


55-e 


41 


35.2 


21.1 


01 


86.6 


51.9 


41 


34.1 


2I.B 


101 


84.0 


j;h.i 


42 


36.0 


21.6 


02 


87.5 


B2.4 


42 


34.9 


23.3 




B4,B 


6a-7 


43 


36.9 


22.1 






62.9 




3S.8 


23.9 




66,6 


67.2 


44 


37.7 


22.6 


04 


89.2 


63.5 


44 


36.6 


24.4 




86.B 


S7.8 


45 


38.6 


23.1 


05 


90.1 


64,0 


45 


37.4 


25.0 




87.3 


68-3 


46 


39.5 


23.6 


106 


90,9 


64.5 


46 


38.2 


26.6 


106 


98,1 


56.9 


47 


40.3 




107 


91.9 


55.0 


47 




26.1 


107 


89.0 


69,4 


48 


41.2 


24.7 




92.6 




48 




26.7 


Ite 


39.3 


60.0 


49 


42.0 


25.2 


109 


93.6 


66'0 


49 


iO'.l 


27.2 


109 


90.6 


60.6 


60 


42.9 


25.7 


110 


94.3 


66.6 


60 


41.6 


27.3 


110 


91.5 


61.1 


51 


43.7 


26,2 




95.2 


57.1 


61 


42.4 


26.3 


1 t 


92.3 


61.7 


62 


44.6 


28.7 


112 


96.1 


67.6 


B2 


43.2 


26.9 


1 2 


93,1 


62.2 


53 


45.5 


27.2 


113 


96.9 


sai 


63 


44.1 


29.4 


1 3 


94.0 


62. B 


54 


46.3 


27,B 


114 


97.6 


58,6 


64 


44.9 


30,0 


1 4 


94,8 




65 


4T.2 


23,3 


115 


98.6 


69.1 


55 


45.7 


30.6 


315 9S,6 


63.9 


66 


49.0 


219 


116 


99,5 


696 


66 


46.6 


31.1 


116 !96.4 


644 


67 


43.9 


29.3 


117 


100.4 


60.1 


£7 


47.4 


31.7 


117 


97.3 


65.0 


6a 


49,7 


29.8 


118 


101.2 


60.7 


58 


48.2 


32.2 


118 


96.1 


65-6 


B9 


50.6 


30.3 


119 


102. 1 


61.2 


£9 


49.1 


32.8 


'•S 


98.9 


661 


60 


61,5 


30.8 


120 


102.9 


61,7 


60 


49,9 


33.3 


120 


99.6 


66.t 


till. 


d.p. 


lu. 


dill. 


^. 


Ul. 


dtiL 


rf.>. 


(«i. 


jfiii. 


dv. 


Int. 


_ 


torB 


PoiQ 


.. 




loi 6 Poirl 







TRAVERSE TABLE. 



D{f, qflat. 4^ dep.forSk Point, \ 


Dif. qflat, 4'dep,fbr2k PoinU. [T.iir. | 


auu 


l«i. 


dtp. 


duL 


laL 


dep. 


di9t. 


laU 


dep. 


diat. 


lut. dep, 1 


1 


oas 


00.6 


61 


49.0 


36.3 


1 


00.8 


00.6 


61 


47.1 


38.7 


2 


01.6 


01.2 


62 


49.8 


36.9 


2 


01.5 


01.3 


62 


47.9 


39.3 


3 


02.4 


01.8 


63 


50.6 


37.5 


3 


02.3 


01.9 


63 


48.7 


40.0 


4 


03.2 


02.4 


64 


51.4 


3ai 


4 


03.1 


02.5 


64 


49.6 


40.6 


6 


04.0 


0.30 


65 


52.2 


3a7 


5 


03.9 


03.2 


66 


60.2 


41.2 


6 


04.8 


03.6 


66 


53.0 


39.3 


6 


04.6 


03.8 


66 


51.0 


41.9 


7 


05.6 


04.2 


67 


53.8 


39.9 


7 


05.4 


04.4 


67 


51.8 


42.5 


6 


06.4 


04.8 


68 


64.6 


40.5 


8 


06.2 


05.1 


68 


52.6 


43.1 


9 


07.2 


05.4 


69 


55.4 


41.1 


9 


07.0 


05.7 


69 


53.3 


43.8 


10 


06.0 


06.0 


70 


56.2 


41.7 


10 


07.7 


06.3 


70 


54.1 


44.4 


11 


06.8 


06.6 


71 


57.0 


42.3 


11 


08.5 


07.0 


71 


54.9 


4M 


12 


09.6 


07.1 


72 


57.8 


42.9 


12 


09.3 


07.6 


72 


56.7 


4B,7 


13 


10.4 


07.7 


73 


58.6 


43.5 


13 


10.1 


08.2 


73 


56.4 


46.3 


14 


11.2 


oas 


74 


59.4 


44.1 


14 


10.8 


06.9 


74 


67.2 


46.9 


15 


12.0 


08.9 


75 


60.2 


44.7 


15 


11.6 


09.5 


75 


68.0 


47.6 


16 


12.8 


09.5 


76 


61.0 


45.3 


16 


12.4 


10.1 


76 


58.7 


4a2 


17 


13.7 


10.1 


77 


61.6 


45.9 


17 


18.1 


10.8 


77 


59.5 


48.8 


18 


14.5 


10.7 


78 


62.6 


46.5 


18 


13.9 


11.4 


78 


60.3 


49.5 


19 


15.3 


11.3 


79 


63.4 


47.1 


19 


14.7 


12.0 


79 


61.1 


50.1 


20 


16.1 


11.9 


80 


64.3 


47.7 


20 


15.5 


12.7 


80 


61.8 


50.7 


21 


16.9 


12,5 


81 


65.1 


48.3 


21 


16.2 


13.3 


61 


62.6 


51.4 


22 


17.7 


13.1 


82 


65.9 


4a9 


22 


17.0 


14.0 


82 


63.4 


52.0 


23 


ia5 


13.7 


83 


66.7 


49.4 


23 


17.8 


14.6 


83 


64.2 


52.7 


24 


19.3 


14.3 


84 


67.5 


50.0 


24 


ia5 


15.2 


84 


64.9 


53.3 


25 


20.1 


14.9 


85 


6a3 


50.6 


25 


19.3 


15.9 


85 


657 


53.9 


26 


20.9 


15.5 


86 


69.1 


51.2 


26 


20.1 


16.5 


86 


66.5 


54.6 


27 


21.7 


16.1 


87 


69.9 


51.6 


27 


20.9 


17.1 


87 


67.2 


65.2 


28 


22.5 


16.7 


88 


70.7 


52.4 


28 


21.6 


17.8 


88 


680 


55.6 


29 


23.3 


17.3 


89 


71.6 


53.0 


29 


22.4 


18.4 


89 


68.8 


56.6 


30 


24.1 


17.9 


90 


72.3 


53.6 


30 


23.2 


19.0 


90 


69.6 


57.1 


31 


24.9 


16.5 


91 


73.1 


64.2 


31 


24.0 


19.7 


91 


70.3 


57.7 


32 


25.7 


19.1 


92 


73.9 


64.8 


32 


24.7 


20.3 


92 


71.1 


58.4 


33 


26.5 


19.7 


93 


74.7 


55.4 


33 


25.5 


20.9 


93 


71.9 


59.0 


34 


27.3 


20.3 


94 


75.5 


56.0 


34 


26.3 


21.6 


94 


72.7 


59.6 


36 


28.1 


20.9 


95 


76.3 


56.6 


35 


27.1 


22.2 


96 


73.4 


60.3 


36 


28.9 


21.4 


96 


77.1 


57.2 


36 


27.8 


24.8 


96 


74.2 


60.9 


37 


29.7 


22.0 


97 


77.9 


57.8 


37 


28.6 


23.5 


97 


75.0 


61.5 


38 


30.5 


22.6 


98 


7a7 


58.4 


38 


29.4 


24.1 


98 


75.7 


6-2.2 


39 


31.3 


23.2 


99 


79.5 


59.0 


39 


30.1 


24.7 


99 


76.5 


62.8 


40 


32.1 


23.8 


100 


80.3 


59.6 


40 


30.9 


25.4 


100 


77.3 


63.4 


41 


32.9 


24.4 


101 


81.1 


60.2 


41 


31.7 


26.0 


101 


7a 1 


64.1 


42 


33.7 


25.0 


102 


81.9 


60.8 


42 


32.5 


26.6 


102 


7a8 


64.7 


43 


34.5 


25.6 


103. 


82.7 


61.4 


43 


33.2 


27.3 


103 


79.6 


65.3 


44 


35.3 


26.2 


104 


83.5 


62.0 


44 


34.0 


27.9 


104 


80.4 


66.0 


45 


36.1 


26.8 


105 


84.3 


62.6 


45 


34.8 


2a5 


105 


81.2 


66.6 


46 


36.9 


27.4 


106 


85.1 


63.1 


46 


35.6 


29.2 


106 


81.9 


67.2 


47 


37.7 


28.0 


107 


85.9 


63.7 


47 


36.3 


29.8 


107 


82.7 


67.9 


48 


38.6 


28.6 


108 


86.7 


64.8 


48 


37.1 


30.4 


108 


83.5 


6a5 


49 


39.4 


29.2 


109 


87.5 


64.9 


49 


37.9 


31.1 


109 


84.3 


69.1 


50 


40.2 


29.8 


110 


88.4 


65.5 


50 


38.6 


31.7 


110 


85.0 


69.6 


51 


41.0 


30.4 


111 


89.2 


66.1 


51 


39.4 


32.3 


111 


85.6 


70.4 


52 


41.8 


31.0 


112 


90.0 


66.7 


52 


40.2 


33.0 


112 


86.6 


71.0 


53 


42.6 


31.6 


113 


90.8 


67.3- 


53 


41.0 


33.6 


U3 


87.3 


71.7 


64 


43.4 


32.2 


114 


91.6 


67.9. 


54 


41.7 


34.3 


114 


86.1 


72.3 


55 


44.2 


32.8 


115 


92.4 


6a5 


55 


42.5 


34.9 


115 


86.9 


73.0 


56 


45.0 


33.4 


116 


93.2 


69.1 


56 


43.3 


35.5 


116 


89.7 


73.6 


57 


45.8 


34.0 


117 


94.0 


69.7 


57 


44.1 


36.2 


117 


90.4 


74.2 


58 


46.6 


34.6 


118 


94.8 


70.3 


58 


44.8 


36.8 


118 


91.2 


74.9 


59 


47.4 


35.1 


119 


95.6 


70.9 


59 


45.6 


37.4 


119 


92.0 


75.5 


60 


43.2 


35.7 


120 


96:4 


71.5 


60 


46.4 


3ai 


120 


92.8 


76.1 


i,%au 


dip. 


laL 


disU 


d^. 


loti 


dUt, 


dep. 


lat. 


dit. 


d€p. 


UL 


for 41 


Points. 








for 4} 


^oints. 1 



TRAVERSE TABLE. 



■1^/. 


Tn^r 


4-d^.for3lPoin, 


l*i/.o 




dtp. 






."'P- 


iliiil.^ la 


00.7 


O0.7 


61 


45.2 


41.0 




00 


Ol.S 


01.3 


62 


45.9 






01 


02.2 


02.0 


63 


46.7 


42^3 




02 


03.0 


02.7 


64 


47.4 


43,0 




03 


03.7 


03.4 




43.2 


43-6 




03 


04.4 


04.0 


66 


4a9 


44.3 




04 


06.a 


04.T 


67 


49.6 


45.0 




04 


06.9 


0S.4 


69 


50,4 


45.7 




05 


06.7 


oe.0 




51.1 


46,3 




06 


07.4 


06.7 


70 


51.9 


47.0 




07 


06.3 


07.4 


71 


52.6 


47.7 




07 


09.9 


09.1 


72 


ra.3 


4a3 




09 


09.G 


03.7 


73 




49.0 




09 


10.4 


D9.4 


74 


Bia 


49.T 




09 


11.1 


10.1 


76 


56.6 


50.4 




10 


11.9 


10.7 


76 


56.3 


51.0 




U 


13.6 


11.4 


77 


57.0 


61.7 




12 


13.3 


12.1 


7B 


67.8 


B2.4 






14.1 


ia.a 


79 


6a5 


B3.0 




13 


14.8 


13.4 


80 


69.3 


63.7 


20 


14 


16.S 


14.1 


ei 


G0.0 


64.4 


31 


It 


■16-3 


14.8 


83 


60.B 


56.1 


22 


IS 


17.0 


15.4 


83 


61.6 


B6.7 


23 


16 


17.8 


16.1 


M 


62.2 


56.4 




17 


lae 


16.B 


BB 


63.0 


67.1 


2S 




19.3 


17.5 




63.7 


B7.7 


26 


IS 


Z0.0 


lai 


87 


64.5 


53,4 




19 


20.7 


lao 




65.2 


69.1 




19 


21.5 


19.6 




65.9 




29 




22.2 


20.1 


90 


66.7 


60.4 


30 


21 


23.0 


20.a 


91 


67.4 


61.1 


31 


21 


23.7 


21.5 


92 


6B.2 


61.8 


32 


22 


24.4 


22.2 


93 


68.9 


62.4 


33 


23 


25.2 


22,8 


94 




63.1 


34 


24 


2B.g 


23.5 


95 


70U 




35 


24 


28.7 


24.2 


96 


71.1 


6416 


36 


25 


27.4 




97 


71,9 


65.1 


37 


26. 


28.2 


2S.B 


93 


72.6 


65.8 




26. 




2e.2 


99 


73,3 


66.5 


39 


27. 


29!6 


26.9 


100 


74.1 


67.2 


40 


2a 


30.4 


27.5 


101 


74.3 


67.8 


41 


29- 


31.1 


m2 


103 


75.6 




42 


29. 


31.9 


28.9 


103 


76.3 




43 


30. 


32-6 




104 


77.1 


69.8 




31, 


33.3 


30:3 


IDS 


77.8 


70-5 




31. 


34.1 


30.9 


106 


7a5 


71.2 


46 


32. 


34.8 


31.6 


107 


79.3 


71-8 


47 




36.8 


32.2 


lOB 


80.0 


72.5 


48 




36,3 


32.9 


109 


80.S 


73-2 


49 


34; 


37.0 


33.6 


110 


81,6 


73.9 


60 


36, 


37. B 


34.3 


111 


82.2 


74.5 


Bl 


36. 


aas 


34.9 


112 


83.0 


75.S 


62 


3& 


39.3 


35.6 


113 


83.7 


75.9 


53 


37. 


40.0 


36.3 


114 


B4.6 


76,5 




38. 


40.7 




115 


B5.2 


77.2 


5S 


38.9 


41.5 


37;6 


116 


8S-9 




56 


39.6 


42.2 


38.3 117 


86.7 


7b!6 


57 


40. 


43.0 


3S.9 lie 


37.4 


79.2 




41.1 


43.7 


39.6 1J9 


88,2 


79.9 






44.5 


40.3 120 


38-9 


B0.6 


60 


42!. 


d.p. 


Inl. 


ii.,A 


d,p. 


(»i. 


*.r, 


rf<p. 



<IL >!■ 


dep./o 


dtp. 




mi 


61 


014 


62 










(Hfi 




047 


66 






115 7 




IKi4 


69 


07.1 


70 


07R 


^t , 


08.5 


72 



01? 


09.Z 










10.fi 

in 


10,6 
!1.3 




12.fl 

13'^ 

14.1 

\5.e 

16.3 
17.0 

is!4 

19,1 
19,6 
20.6 
31.2 
21.9 
23.6 
23.3 
21.0 
24,7 
25.5 
26.2 
26.9 
27.6 
2a3 
29-0 
29.7 
30.4 

III 

W,5 


12 

12 7 

13 4 

14 1 
14 8 
IBS 

16 3 
ITO 

17 7 
19 4 
111 

206 
212 
21.9 
22.6 
E3.3 
24.0 
247 
25.5 

269 
37.6 
2B.3 
29.0 
29.7 
30.4 
31.1 
31.8 
32.5 


1 


33.2 
33.9 

4,6 
36,4 
36.1 
3&B 

7.5 
J8.2 

8.9 

o!3 

1.0 
2!4 


33.2 
33.9 
34.6 
35.4 
36.1 
36.9 
37,5 

39;g 
40.3 
41-0 
41.7 
4^4 
1,1. 
or 4P 


J 

] 
J 

1 
1 
1 
1 

1 
1 
1 
1 



84 



TABLE V. 



A TABLE OP RUMBS, 



"■ »;. 



8H0WIN0 



THE DEGREES, MINUTES, AND SECONDS, THAT EVERY POINT AND 
QUARTER-POINT OF THE COMPASS MAKES WITH THE MERIDIAN. 



NORTH. 



N»E.byN. 



N. by E. 



N.N.E. 



N.E. 



N.E by E. 



E.N.E 



E. by N. 



N. by W. 



N.N.W. 



N.W.byN. 



N.W. 



N.W.byW. 



W.N.W. 



W. by N. 



East. I West. 



Pts. 


qr. 





/ 


// 


Pts. 


qr. 





1 


2 


48 


45 





1 





2 


5 


37 


30 





2 





3 


8 


26 


15 





3 


1 





11 


15 





1 





1 


1 


14 


3 


45 


1 


1 


1 


2 


16 


52 


30 


1 


2 


1 


3 


19 


41 


15 


1 


3 


2 





22 


30 





2 





2 


1 


25 


18 


45 


2 


1 


2 


2128 


7 


30 


2 


2 


2 


3|30 


56 


15 


2 


3 


3 





33 


45 





3 





3 


1 


36 


33 


45 


3 


1 


3 


2i39 


22 


30 


3 


2 


3 


3 


42^ 


11 


15 


3 


3 


4 





45 








4 





4 


1 


47 


48 


45 


4 


1 


4 


2 


50 


37 


30! 4 


2 


4 


3 153 


26 


15 


4 


3 


5 





56 


15 





5 





5 


1 


59 


3 


45 


5 


1 


5 


2 


61 


52 


30 


5 


2 


5 


3 64 


41 


15 


5 


3 


6 


67 


30 





6 





6 


1 


70 


18 


45 6 


1 


6 


2 


73 


7 


30 6 


2 


6 


3 


75 


56 


15 6 . 


3 


7 





78 


45 





7 





7 


1 81 


33 


45 17 


1 


7 


2 


84 


22 


30 


7 


2 


7 


3 


87 


11 


15 


7 


3 


8 





90 








8 






SOUTH. 



S. by E. 



o.o.E. 



S.E. by S. 



S. by W. 



S.S.W. 



S.E. 



S.W.byS. 



S.W. 



S.E. by E. S.W.byW. 



E. ib.E. 



E. by S. 



East. 



W.S.W. 



W. by S. 



West. 



^ 



WORKMAN'S TABLE, 



FOR CORRECTING THE MIDDLE LATITUDE. 



86 



PIFFFERENCE OF LATITUDB. 



[Table YL 



Bfid. 
Lat. 

o 

15 
16 
17 
18 
19 
20 
21 
22 
23 
24 

25 
26 
27 
28 
29 
30 
31 
32 
33 
34 

35 
36 
37 
38 
39 
40 
41 
42 
43 
44 

46 
46 
47 
48 
49 
50 
51 
52 
53 
54 

55 
56 
57 
58 
59 
60 
61 
62 
63 
64 

65 
66 
67 
68 
69 
70 
71 
72 



30 



o 


































































02 
02 
02 
02 
02 
02 
02 
02 
02 
02 

02 
02 
02 
02 
02 
02 
02 
02 
02 
02 

02 
02 
02 
02 
02 
02 
02 
02 
02 
02 

02 
02 
02 
02 
02 
02 
02 
02 
02 
02 

02 
02 
02 
02 
02 
02 
02 
02 
02 
02 

02 
02 
02 
02 
02 
03 
04 
04 



40 

o / 

03 
03 
03 
03 
03 











03 
03 
03 
03 
03 



03 
03 
03 
03 
03 
03 
03 
03 
03 
03 

03 
03 
03 
03 
03 
03 
03 
03 
03 







03 








03 
03 



03 










03 
03 
03 
03 
03 
03 
03 

03 
03 
03 









03 
03 
03 
03 
03 
004 
04 

04 
04 
04 
04 
05 
05 
06 
06 



04 
04 
04 
04 
04 
04 
04 
04 
04 
04 






















04 
04 
04 
04 
04 
04 
04 
04 
04 
04 



04 
004 
04 
04 
04 
04 
04 
04 
004 
04 

04 
04 
04 
04 
04 
04 
04 
04 
04 
04 


















04 
04 
04 
04 
04 
04 



05 









05 
05 
06 



06 
06 
006 
06 
06 
06 
07 
08 










60 



o 













06 
06 
06 
06 
06 
06 
06 
06 
06 
06 



05 
05 
05 
05 
05 
05 
05 
05 
05 
05 










05 
05 
05 
05 
05 
05 
05 
05 
05 








05 

05 
05 
05 
05 
05 
05 
05 
05 
06 











0.06 

006 
006 
06 
006 
006 
06 
07 
07 
07 
08 

008 
08 
008 
08 
09 
09 
09 
10 



70 



o 























09 
09 
06 
06 
07 
07 
07 
07 
07 
07 

07 
07 
07 
06 
06 
06 
06 
06 
06 



006 
















06 
06 
06 
06 
06 
06 
06 
06 
07 
07 



07 
07 
07 
07 
07 
07 
07 
07 
07 
08 















08 
08 
08 
09 
09 
09 





09 
09 
09 
09 



10 
10 
11 
11 
12 
13 
13 
14 



8° 


/ 


12 


12 


11 


11 


10 


09 


009 


09 


009 


09 


09 


009 


08 


008 


08 


08 


08 


08 


08 


08 


08 


08 


08 


08 


008 


08 


08 


008 


009 


09 


09 


09 


09 


09 


09 


09 


09 


09 


09 


10 


10 


10 


11 


11 


12 


12 


12 


12 


13 


13 


13 


14 


15 


15 


16 


17 


18 


19 



90 

O / 

15 
15 
14 
14 

































13 
12 
12 
12 
12 
11 

11 
11 
11 
10 
10 
10 



10 
10 
10 
10 



10 
10 
10 
10 
10 



10 
10 



10 

11 
11 



11 
11 
11 
11 
11 
11 
11 
12 
12 
13 

13 
13 
14 
14 
15 
15 
15 
16 
16 
17 

17 
18 
18 
19 
20 
21 
22 
23 



100 



o 




































l'9 
18 
17 
17 
16 
16 
15 
15 
15 
14 



14 
14 
14 
13 
13 
13 
13 
13 
13 
13 



13 
13 
13 
13 
13 
13 
13 
13 
14 



14 



14 
14 
14 
14 
14 
14 
14 
15 



16 
16 

16 
16 
17 
17 
18 
19 
19 
20 
20 
21 

21 
22 
23 
24 
25 
26 
27 
29 



IP 

o / 
23 
22 
21 
20 
19 
18 
&18 
OIT 
f 17 
16 

16 
16 
16 
15 
15 
15 
15 
15 
15 
15 































15 
15 
15 
15 
15 
15 
15 
15 



16 
16 












16 
16 
16 
16 
17 
17 



17 
18 
18 
19 



19 
20 
20 
21 
22 



023 
23 



24 
24 
25 

25 
26 



27 
28 
30 
31 








33 
35 



tmm 



i*iftfMMaa^ 



H»*a«a 



■H 



tm 



87 



liable yl] 




niPFERENCE OF LATITCDE. 








Mid. 
Lat. 


12=> 


130 


140 


15^ 


16=> 


170 


18^ 


19=» 


20^ 


o 


O / 


/ 


/ 


/ 


/ 


/ 


/ 


/ 


/ 


15 


27 


31 


35 


40 


45 


51 


53 


1 06 


1 14 


16 


26 


30 


34 


33 


43 


49 


56 


1 03 


1 11 


17 


25 


28 


32 


37 


42 


43 


54 


1 01 


1 08 


18 


24 


27 


31 


36 


41 


46 


52 


58 


1 06 


19 


23 


026 


30 


34 


40 


45 


50 


56 


1 03 


20 


22 


025 


29 


33 


38 


43 


43 


054 


1 00 


21 


21 


25 


29 


33 


37 


42 


47 


53 


58 


22 


20 


24 


28 


32 


36 


41 


46 


51 


56 


23 


20 


24 


28 


32 


36 


40 


45 


50 


55 


24 


19 


23 


27 


31 


q 35 


39 


44 


48 


53 


25 


19 


23 


27 


31 


35 


39 


43 


47 


52 


26 


19 


22 


26 


30 


34 


38 


42 


47 


52 


27 


19 


22 


26 


30 


33 


38 


42 


46 


51 


28 


13 


21 


25 


29 


33 


37 


41 


46 


51 


29 


18 


21 


25 


29 


32 


36 


41 


45 


050 


30 


18 


21 


25 


28 


32 


36 


41 


45 


050 


31 


18 


21 


25 


28 


32 


36 


41 


45 


50 


32 


18 


21 


25 


28 


31 


36 


41 


45 


050 


33 


18 


21 


24 


27 


31 


35 


40 


44 


49 


34 


18 


21 


24 


27 


31 


35 


40 


44 


49 


35 


18 


21 


24 


27 


31 


35 


40 


44 


49 


36 


18 


21 


24 


27 


31 


35 


40 


44 


49 


37 


18 


21 


24 


27 


31 


35 


40 


44 


49 


38 


18 


21 


24 


27 


31 


36 


40 


45 


50 


39 


18 


21 


25 


28 


32 


36 


41 


45 


50 


40 


18 


22 


25 


28 


32 


36 


41 


45 


50 


41 


18 


22 


25 


28 


32 


37 


41 


45 


50 


42 


18 


22 


26 


29 


33 


37 


42 


46 


51 


43 


19 


23 


26 


30 


34 


38 


42 


46 


51 


44 


19 


23 


27 


30 


34 


38 


43 


0'47 


52 


45 


19 


23 


27 


31 


35 


39 


43 


47 


52 


46 


19 


23 


27 


31 


35 


39 


44 


48 


53 


47 


20 


23 


27 


31 


35 


40 


44 


49 


54 


48 


20 


23 


27 


31 


35 


40 


45 


50 


55 


49 


21 


24 


28 


32 


36 


41 


45 


51 


57 


50 


21 


24 


28 


32 


36 


41 


46 


52 


58 


51 


21 


24 


28 


32 


37 


42 


47 


53 


59 


52 


22 


25 


29 


33 


37 


42 


48 


54 


1 00 


53 


22 


25 


29 


33 


38 


43 


49 


55 


1 01 


54 


23 


26 


30 


34 


39 


44 


50 


56 


1 02 


55 


23 


26 


30 


35 


40 


45 


51 


57 


1 03 


56 


24 


27 


31 


36 


41 


46 


52 


58 


1 04 


57 


24 


28 


32 


37 


42 


43 


54 


1 00 


1 06 


58 


25 


29 


33 


38 


44 


50 


55 


1 02 


1 08 


59 


26 


30 


34 


39 


45 


51 


57 


1 04 


1 10 


60 


27 


31 


35 


40 


46 


52 


59 


1 06 


1 13 


61 


27 


31 


36 


41 


47 


54 


1 01 


1 08 


1 15 


62 


28 


32 


37 


42 


49 


56 


1 03 


1 10 


1 18 


63 


29 


33 


39 


44 


51 


58 


1 05 


1 12 


1 21 


64 


29 


34 


40 


46 


53 


1 00 


1 07 


1 14 


1 24 


65 


30 


35 


41 


48 


55 


1 02 


1 09 


1 17 


1 27 


66 


31 


37 


43 


50 


58 


1 05 


1 12 


1 21 


1 31 


67 


33 


38 


45 


53 


1 00 


1 07 


1 16 


1 25 


1 35 


68 


34 


40 


48 


55 


1 02 


1 10 


1 19 


1 30 


1 39 


69 


36 


42 


50 


58 


1 05 


1 13 


] 23 


1 34 


1 44 


70 


38 


44 


52 


1 00 


1 08 


1 17 


1 28 


1 39 


1 50 


71 


40 


46 


55 


1 03 


1 12 


1 22 


1 32 


1 44 


1 56 


72 


42 


49 


058 


1 06 


1 16 


1 27 


1 38 


1 50 


2 04 



TABLE OF MERIDIONAL PARTS. 



^ 



tkbht vit. 



91 



T 


Oo 


lo 


2o 


30 


40 


50 


60 


70 


SO 


90 


100 


IP 


120 


130 








60 


120 


180 


240 


300 


361 


421 


482 


542 


603 


664 


725 


787 


1 


1 


61 


121 


181 


241 


301 


362 


422 


483 


543 


604 


665 


726 


788 


2 


2 


62 


122 


182 


242 


302 


363 


423 


484 


544 


606 


666 


727 


789 


3 


3 


63 


123 


183 


243 


303 


364 


424 


485 


545 


606 


667 


728 


790 


4 


4 


64 


124 


184 


244 


304 


366 


426 


486 


646 


607 


668 


729 


791 


6 


5 


65 


125 


186 


246 


306 




426 


487 


547 


608 


669 


730 


792 


6 


6 


66 


126 


186 


246 


306 


367 


427 


488 


548 


609 


670 


731 


793 


7 


7 


67 


127 


187 


247 


307 


368 


428 


489 


549 


610 


671 


732 


794 


8 


8 


68 


128 


188 


248 


308 


369 


429 


490 


560 


611 


672 


734 


796 


9 


9 


69 


129 


189 


249 


309 


370 


430 


491 


551 


612 


673 


736 


796 


10 


10 


70 


130 


190 


250 


310 


371 


431 


492 


662 


613 


674 


736 


797 


11 


11 


71 


131 


191 


261 


311 


372 


432 


493 


653 


614 


676 


737 


796 


12 


12 


72 


132 


192 


252 


312 


373 


433 


494 


654 


615 


676 


738 


799 


13 


13 


73 


133 


193 


263 


313 


374 


434 


495 


555 


616 


677 


739 


800 


14 


14 


74 


134 


194 


254 


314 


376 


435 


496 


566 


617 


678 


740 


801 


15 


15 


75 


135 


196 


266 


316 


376 


436 


497 


557 


618 


679 


741 


802 


16 


16 


76 


136 


196 


266 


316 


377 


437 


498 


558 


619 


680 


742 


803 


17 


17 


77 


137 


197 


267 


317 


378 


438 


499 


559 


620 


681 


743 


804 


18 


18 


78 


138 


198 


258 


318 


379 


439 


500 


660 


621 


682 


744 


805 


19 


19 


79 


139 


199 


259 


319 


380 440 


601 


561 


622 


683 


745 


806 


20 


20 


80 


140 


200 


260 


320 


381 


441 


502 


562 


623 


684 


746 


807 


21 


21 


81 


141 


201 


261 


321 


382 442 


503 


563 


624 


686 


747 


806 


22 


22 


82 


142 


202 


262 


322 


383 443 


604 


564 


626 


687 


748 


809 


23 


23 


83 


143 


203 


263 


323 


384 


444 


605 


565 


626 


688 


749 


810 


24 


24 


84 


144 


204 


264 


324 


385 


446 


506 


567 


627 


689 


750 


811 


25 


25 


85 


145 


205 


266 


325 


'386 


446 


507 


568 


628 


690 


761 


812 


26 


26 


86 


146 


206 


266 


326 


387 


447 


508 


569 


629 


691 


762 


813 


27 


27 


87 


147 


207 


267 


327 


388 448 


509 


670 


631 


692 


753 


815 


28 


28 


88 


148 


208 


268 


328 


389 449 


610 


571 


632 


693 


754 


816 


29 


29 


89 


149 


209 


269 


330 


390 450 


611 


672 


633 


694 


755 


817 


30 


30 


90 


150 


210 


270 


331 


391 


461 


612 


673 


634 


696 


756 


818 


31 


31 


91 


151 


211 


271 


332 


392 


452 


613 


674 


636 


696 


757 


819 


32 


32 


92 


152 


212 


272 


333 


393 


453 


514 


675 


636 


697 


758 


820 


33 


33 


93 


163 


213 


273 


334 


394 


454 


616 


576 


637 


698 


759 


821 


34 


34 


94 


164 


214 


274 


335 


395 


456 


616 


577 


638 


699 


760 


822 


35 


35 


95 


155 


216 


276 


336 


396 


466 


517 


578 


639 


700 


761 


823 


36 


36 


96 


166 


216 


276 


337 


397 


457 


518 


579 


640 


701 


762 


824 


37 


37 


97 


167 


217 


277 


338 


398 


468 


619 


680 


641 


702 


763 


826 


38 


38 


98 


168 


218 


278 


339 


399 


469 


520 


581 


642 


703 


764 


826 


39 


39 


99 


169 


219 


279 


340 


400 


460 


521 


682 


643 


704 


765 


827 


40 


40 


100 


160 


220 


280 


341 


401 


461 


522 


583 


644 


705 


766 


828 


41 


41 


101 


161 


221 


281 


342 


402 


462 


523 


584 


646 


706 


767 


829 


42 


42 


102 


162 


222 


282 


343 


403 


463 


624 


585 


646 


707 


768 


830 


43 


43 


103 


163 


223 


283 


344 


404 


464 


626 


586 


647 


708 


769 


831 


44 


44 


104 


164 


224 


284 


346 


405 466 


626 


587 


648 


709 


770 


832 


45 


46 


106 


166 


225 


286 


346 


406 466 


627 


588 


649 


710 


771 


833 


46 


46 


106 


166 


226 


286 


347 


407 


467 


528 


589 


660 


711 


772 


834 


47 


47 


107 


167 


227 


287 


348 


408 


468 


529 


590 


651 


712 


773 


a36 


48 


48 


108 


168 


228 


288 


349 


409 


469 


530 


591 


652 


713 


774 


836 


49 


49 


109 


169 


229 


289 


350 


410 470| 


531 


592 


663 


714 


775 


837 


60 


50 


110 


170 


230 


290 


351 


411 


471 


632 


593 


654 


715 


T77 


838 


51 


51 


111 


171 


231 


291 


352 


412 472 


633 


594 


665 


716 


778 


839 


52 


52 


112 


172 


232 


292 


353 


413 473 


634 


595 


656 


717 


779 


849 


63 


53 


113 


173 


233 


293 


354 


414 474 


535 


596 


657 


718 


780 


841 


54 


54 


114 


174 


234 


294 


356 


415 476 


536 


597 


658 


T19 


m 


842 


55 


55 


116 


175 


235 


296 


356 


416 


477 


637 


598 


659 


7^ 


782 


843 


56 


56 


116 


176 


236 


296 


357 


417 


478 


538 


699 


660 


^ 


783 


844 


57 


67 


117 


177 


237 


297 


358 


418 479 


639 


600 


661 




84S 


58 


58 


118 


178 


238 


298 


359 


419 480 


540 


601 


662 


7?3 


m 

798 


846 


59 


59 


119 


179 


239 


299 


360 


420 


4811 


541 


602 


663 


724 


847 



VMP 



TABLE Ttl. 



M. 


14^ 16° 16°j U' 


18= 


19° 


30^ 


31=1 32" 


23° 24° 


25° 


^ 


37= 





S4S 910' ST3 1035 


1099 


1161 


1225 


12891364 


1410 1484 


1550 


1616 


1684 


1 


BBO; 911 974 


36 


99 


63 


26 


90 




20 


85 


61 




85 


1 


851; 913, 975 


37 


1100 


G4 


27 


91 




3 


86 


53 




86 


3 


062: 914, 976 


38 


01 


65 2\ 


92 






87 


fi3 




87 


4 


853 9151 977 


M 


02 


6« 2; 


93 




2: 


88 


64 


21 


88 


B 


864 916, 978 


41 


03 


67 


301 95 




24 


91 


G{ 








8551 917. 979 




05 


68 


32; 96 




25 


91 






90 


7 


8561 918' 980 


4: 


06 


69 


33! 97 




26 


92 


K 




91 


a 


867, 919 


981 


44 


07 


70 


341 98 




27 


93 


59 




93 


9 


8GB| 920 


98-, 


46 


Od 


71 


35 99 




38 


94 


60 




94 





859. 921 


9K 


1946 


1109 


1172 


1236! 1300 




1430 


1495 


1661 




I69B 




B60 922 




47 


10 


73 


37 01 




31 


96 


62 




96 


3 


861 923 


981 




11 


74 




02 


67 


32 


97 


63 




97 


3 


8H2 924 


9B6; 49 


12 


75 






6 


33 


98 


64 


31 




i 


863 9'i5 


987 


50 


]3 


7( 




fy. 




34 


99 


65 




99 


s 


B64; 926 


98f 


51 


14 




4) 


05 




35 


150O 


67 


3! 


1700 


& 


866 927 




62 


15 


78 




Ofi 


7 


36 


02 


68 


34 


01 


7 


B66 02S 


9il( 


53 


16 


79 


43 


07 




37 


OJ 


69 


35 


03 


9 


B671 929 


Ml 


M 


17 


81 


44 


03 




38 


04 


70 


37 


04 


19 


868 H30| 993 


65 


18 






10 


74 


39 


05 


:i 


38 


03 


20 


869, 93! 


994 


1056 


1119 


1183 




1311 


1375 


144( 


1E06 


1572 


1639 


1706 


21 


870, 93! 


995 


hi 


20 


84 




13 


70 




07 


73 


40 


07 


23 


B7ll 933 


996 


5F 


21 


85 




13 


77 


43 


03 


74 


41 


OS 


£3 


872, 934 


997 


59 


22 


86 




4 




44 


09 


76 


42 09 1 


24 


873' 935 


998 


eo 


23 


67 


61 


5 




4£ 


10 


77 


43 It 1 


25 


9741 936 


999 


61 


25 


89 


52 


6 


at 




11 


78 


44 


12 


26 


875; 937; 1000 


63 


21 




63 


1 






13 


19 




13 


27 


a-e. 933 lOOl 


64 




90 54 


19 






14 


80 


47 


14 


28 


877, 939,100-^ 


65 


2i 


91 55 


19 






16 


81 


48 


16 


29 


878 941.1003 


■3G 




92 66 


20 


85J 50 


16 


62 


49 


16 


30 


S79. 942 1004 


1067 


11301193:1257 


1321 


1386 1461 


1617 


1533 


1650 


1717 


31 


880; 913 Oi 


68 


3]| 9t| 58 




871 52 


IS 


£4 


SI 


IS 


34 


882, 944 06 




32 


95[ 59 


2^ 


88, 5: 


19 


83 


52 


20 


33 


883' 945 07 


71 


33 


96 60 


25 


89 56 


21 


Bf 


53 


21 


34 


884 946 08 




34 


98 


61 


26 


90, 66 






64 


22 


35 


883 947 09 


71 


35 


99 


62 


37 


92 67 


Z 


891 66 


23 


36 


886 948 10 




36 


1200 


64. 28 


93 




24 


9Q| 57 


24 


37 


887 919 H 


74 


3' 


01 


651 39 


94 




25 


91; 58 


35 


3S 


383 930 12 


76 




02 


661 30 


93 


CO 


26 


9^ 




26 


39 


K9 951^ 13 


76 


3< 


03 


67| 31 


96 


61 


27 


93 


60 


27 


40 


890. 952 1014 


1077 


1140 


1204 


12681332 


1397 


1462 


1628! 1694 


1661 


1729 


4) 


89i; 953, 15 


78 


41 


03 


69 33 


98 


63 


39 96 


62 


30 


42 


8921 954, 16 


7! 




06 


70 


34 


99 


64 


30j 97 


6; 


31 


43 


m 955 19 




44 


07 


71 


3E 


1400 


65 




64 


33 


44 


B94: 956l 19 


ei 


46 


0^ 


72 







67 


33' 9! 


CE 


33 


4E 


895; 957 iO 


S2 


46 


09 




» 


02 


68 


33 1600 


67 


34 


411 


S96l 958! 21 


B4 


47 


10 


74 


39 


03 


69 


35 


01 




36 




8971 959, 22 


es 


48 


li 


75 


40 


05 


70 


36 


02 


6! 


36 


4B 


8981 960 23 


as 


49 


12 


76 


4i 


06 


71 


37 


03 


70 






899 961 ! 24 




50 




77 


42 


07 


72 


38 


04 


71 


39 


SO 


EOO 962' 1025 


lose 


115 


1216 


1278 


1343 


1403 


1473 


1S39 


1606 


1672 


1740 


Gl 


901 


963 26 


69 


52 


16 


80 


4- 


09 


l 


40 


06 


73 


41 


52 


902 


964 27 


10 


5: 


17 


81 




10 




4 


09 


fB 


42 


E3 


SOJ 


965, 28 


SI 




IS 


82 


41 


11 


7i 


43 


09 


76 


43 


S4 


901 


956 29 fl'i 


5: 


19 




47 


12 


7- 


4; 


10 


77 


44 


55 


90- 


9681 30 93 


66 


20 


84 


49 


13 


79 




11 


78 


46 


ec 


P03 


959 31 94 


57 


1\ 


85 


49 


14 


60 




12 


79 


47 


67 


907 


970! 32 9f 


58 






to 


15 


9 


47 


13 




48 


BS 


903 


<)7i: 33 9f 


59 


23 


ff 


52 


16 


62 


48 


14 


8 


49 


59 


S03 


9721 34 97 


60 


24 




53 


18 


83 


49 


15 




50 















TABLE VII. 










99 


M. 


280 


290 


30O 


310 


320 


1 
330 340 


350, 36° 370 


380 39<5 


40° 


410 





1751 


1819 


1888 


1958 


2028 


2100)2171 


'2244 2318 2393 


2468 2545!2623 


2702 


1 


52 


21 


90 


59 


30 


OH 73 


46. 19 


94 


70 


46 


24 


03 


2 


53 


22 


91 


60 


31 


02, 74 


47 


20 


95 


71 


48 


25 


04 


3 


55 


23 


92 


62 


32 


03 76 


48 


22 


96 


72 


49 


27 


06 


4 


56 


24 


93 


63 


33 


04 76 


49 


23 


98 


73 


EO 


28 


07 


5 


57 


26 


94 


64 


34 


05 


76 


50: 24 


99 


75 


51 


29 


08 


6 


58 


26 


96 


65 


35 


07 


79 


52 


25.2400 


76 


53 


31 


10 


7 


59 


27 


96 


ee 


37 


08 


60 


53 


27 


01 


77 


54 


32 


11 


8 


60 


29 


98 


67 


38 


09 


81 


54 


28 


03 


78 


55 


33 


12 


9 


61 


30 


99 


69 


39 


10 


82 


56 


29 


04 


80 


57 


34 


14 


10 


1762 


1831 


1900 


1970 


2040 


211l|2l84 


2257,2330!2405 


248l'2558'2636 


2716 


11 


64 


32 


01 


71 


41 


13 85 


58, 32 
59^ 33 
60 34 


06 


62 


59 


37 


16 


12 


65 


33 


02 


72 


43 


14 86 


08 


84 


60 


38 


18 


13 


66 


34 


03 


73 


44 


15 


87 


09 


85 


62, 40 


19 


14 


67 


35 


05 


74 


45 


16 


88 


61 35 


10 


86 


63 


41 


20 


16 


68 


37 


06 


76 


46 


17 


SO 


63 37 


11 


87 


64 


42 


22 


16 


69 


38 


07 


77 


47 


19 


91 


64 


38 


13 


69 


66 


44 


23 


17 


70 


39 


08 


78 


48 


20 


92 


65 


39 


14 


90 


67 


45 


24 


18 


72 


40 


09 


79 


50 


21 


93 


66 


40 


15 


91 


68 


46 


26 


19 


73 


41 


10 


80 


61 


22 


94 


68 


42 


16 


92 


69 


48 


27 


20 


1774 


1842 1912 


1981 


2052 


2123 


2196 


2269 2343 


24l8 2494'257l'2649 


2728 


21 


75 


43 


13 


83 


53 


25 


97 


70 44 


19 


95 


72 50 


29 


22 


76 


45 


14 


84 


54 


26 


98 


71 


45 


20 


96 


73 


61 


31 


23 


77 


46 


15 


85 


56 


27 


99 


72 


46 


22 


98 


75 


63 


32 


24 


78 


47 


16 


86 


67 


28 


2200 


74 


48 


23 


99 


76 


54 


33 


25 


60 


48 


17 


87 


58 


29 


02 


75 


49 


24 2500 


77 


55 


36 


26 


81 


49 


18 


88 


59 


31 


03 


76 


50 


25 


01 


78 


57 


36 


27 


82 


50 


20 


90 


60 


32 


04 


77 


51 


27 


03 


60 


58 


37 


28 


83 


62 


21 


91 


61 


33 


05 


79 


53 


28 


04 


81 


59 


39 


29 


84 


53 


22 


92 


63 


34 


07 


60 


54 


29 


05 


82 


61 


40 


30 


1785 


1854* 1923' 1993 


2064 


2135 


2208 


2281 


2355 


24302e06'2584'2662 


2742 


31 


86 


55 


24 


94 


65 


37 


09 


82 


56 


32 08 


85 


63 


43 


32 


87 


56 


25 


95 


66 


38 


10 


83 


58 


33 


09 


66 


65 


44 


33 


89 


57 


27 


97 


67 


39 


11 


85 


69 


34 


10 


88 


66 


46 


34 


90 


68 


28 


98 


69 


40 


13 


86 


60 


35 


12 


89 


67 


47 


35 


91 


60 


29 99 


70 


41 


14 


87 


61 


37 


13 


90 


69 


48 


36 


92 


61 


30 2000 


71 


43 


15 


88 


63 


38 


14 


91 


70 


60 


37 


93 


62 


31 


01 


72 


44 


16 


90 


64 


39 


15 


93 


71 


61 


38 


94 


63 


32 


02 


73 


45 


17 


91 


65 


40 17 


94 


73 


52 


39 


95 


64 


34 


04 


75 


46 


19 


92 


66 


42; 18 


95 


74 


54 


40 


1797 


1866 19352005 


2076 


2147 


2220 


2293 2368 


24432519 26972676 


2756 


41 


98 


66 


36 


06 


77 


49 


21 


95 


69 


44' 21 


98 76 


56 


42 


99 


68 


37 


07 


78 


50 


22 


96 


70 


46| 22 


99 


78 


68 


43 


1800 


69 


38 


09 


79 


51 


24 


97 


71 


47| 23 


2600 


79 


69 


44 


01 


70 


39 


10 


80 


62 


25 


98 


73 


48 24 


02 


60 


60 


45 


02 


71 


41 


11 


8-2 


53 


26 


99 


74 


49 26 


03 


82 


62 


46 


03 


72 


42 


12 


83 


55 


27 


2301 


75 


51 


27 


04 


83 


63 


47 


05 


73 


43 13 


84 


56 


28 


02 


76 


52 28 


06 


84 


64 


48 


06 


75 


44 


14 


85 


57 


30 


03 


78 


53, 30 


07 


86 


66 


49 


07 


76 


45 


15 


86 


58 


31 


04 


79 


64| 31 


08 


87 


67 


50 


1803 


1377 


1946 2017 


2088 


2159 


2232 


2306 


2380 


2456 2532 


2610 2688 


2768 


51 


09 


78 


48, 18 


89 


61 


33 


07 


81 


57| 33 


11 


to 


70 


52 


10 


79 


49 19 


90 


62 


35 


08 


83 


68 35 


12 


91 


71 


53 


11 


80 


50 20 


91 


63 


36 


09 


84 


59 36 


14 


92 


72 


54 


13 


81 


61 21 


92 


64 


37 


11 


85 


61 37 


15 


94 


74 


56 


14 


83 


62 22 


94 


65 


38 


12 


66 


62 38 


16 


95 


76 


56 


15 


84 


63 24 


95 


67 


39 


13 


88 


63 40 


17 


96 


76 


57 


16 


85 


65 25 


96 


68 


41 


14 


89 


64| 41 


19 


98 


78 


68 


17 


86 


66 


26 


97 


69 


42 


16 


90 


66 42 


20 


99 


79 


69 


18 


87 67| 


27 98 


70 431 17' 911 


67j 44 


2127001 


80 



I 



H 



tABLE ViU 



■■ 



t 



M. 


1 
2 
3 
4 
6 
6 
7 
8 
9 

10 
11 
12 
13 
14 
16 
16 
17 
18 
19 

20 
21 
22 
23 
24 
26 
26 
27 
28 
29 

30 
31 
32 
33 
34 
36 
36 
37 
3d 
39 

40 
41 
42 
43 
44 
46 
46 
47 
48 
49 

60 
61 
62 
63 
64 
66 
66 
57 
68 
69 



420 


43° 


2782 


2863 


83 


64 


84 


66 


86 


67 


87 


69 


88 


70 


90 


71 


91 


73 


92 


74 


94 


76 


2796 


2877 


97 


78 


98 


80 


99 


81 


2801 


82 


02 


84 


03 


85 


06 


86 


06 


88 


07 


89 


2809 


2891 


10 


92 


11 


93 


13 


96 


14 


96 


16 


97 


17 


99 


18 


2900 


20 


02 


21 


03 


2822 


2904 


24 


06 


25 


07 


26 


08 


28 


10 


29 


11 


30 


13 


32 


14 


33 


16 


34 


17 


2836 


2918 


37 


19 


39 


21 


40 


22 


41 


24 


43 


25 


44 


26 


46 


28 


47 


29 


48 


31 


2849 


2932 


61 


33 


62 


35 


64 


36 


65 


37 


66 


39 


68 


40 


69 


42 


60 


43 


62 


44 



4401 4501 460 



2946 
47 
49 
60 
61 
63 
64 
66 
67 
68 

2960 
61 
63 
64 
66 
67 
68 
70 
71 
72 



3030 
31 
33 
34 
36 
37 
38 
40 
41 
43 

3044 
46 
47 
48 
60 
61 
63 
64 
65 
67 



2974 


3068 


76 


60 


76 


61 


78 


63 


79 


64 


81 


66 


82 


67 


83 


68 


85 


70 


86 


71 



2988 3073 



89 
91 
92 
93 
95 
96 
98 
99 
3000 

3002 
03 
05 
06 
07 
09 
10 
12 
13 
14 

3016 
17 
19 
20 
21 
23 
24 
26 
27 
29 



74 
76 
77 

78 
80 
81 
83 
84 
85 



3116 
17 
18 
20 
21 
23 
24 
26 
27 
29 

3130 
31 
33 
34 
36 
37 
39 
40 
42 
43 

3144 
46 
47 
49 
60 
62 
63 
66 
66 
67 

3169 
60 
62 
63 
66 
66 
68 
69 
71 
72 



3087 3173 
88 76 



90 
91 
93 
94 
96 
97 
98 



76 

781 

79 

81 

82 

84 

86 



3100 87 

3101 3188 



03 
04 
05 
07 
08 
10 
11 



90 
91 
92 
94 
95 
97 
98 



13 3200 
14i 01 



470 480 



3203 
04 
06 
07 
09 
10 
12 
13 
14 
16 

3217 
19 
20 
22 
23 
26 
26 
28 
29 
31 

3232 
34 
36 
37 
38 
40 
41 
42 
44 
46 

3247 
48 
60 
61 
53 
54 
66 
67 
69 
60 

3262 
63 
65 
66 
68 
69 
71 
72 
74 
76 

3277 
78 
80 
81 
83 
84 
86 
87 
89 
90 



3292 
93 
95 
96 
98 
99 

3301 
02 
03 
05 

3306 
08 
09 
11 
12 
14 
16 
17 
19 
20 

3322 
24 
26 
26 
28 
29 
31 
32 
34 
35 

3337 

38 
40 
41 
43 
44 
46 
47 
49 
60 

3352 
63 
66 
66 
68 
59 
61 
62 
64 
65 

3367 
68 
70 
71 
73 
74 
76 
78 
79 
81 



3382 
84 
85 
87 
88 
90 
91 
93 
94 
96 

3397 
99 

3400 
02 
03 
05 
07 
08 
10 
11 

3413 
14 
16 
17 
19 
20 
22 
23 
25 
27 

3428 
30 
31 
33 
34 
36 
37 
39 
40 
42 

3443 
46 
47 
48 
60 
51 
63 
64 
66 
67 

3469 
60 
62 
64 
65 
67 
68 
70 
71 
73 



600 


61° 


620 


630 


640 


3474 


3669 


3666 


3764 


3866 


76 


70 


67 


66 


67 


78 


72 


68 


67 


68 


79 


74 


70 


09 


70 


81 


75 


72 


70 


71 


82 


77 


73 


72 


73 


84 


78 


76 


74 


75 


86 


80 


77 


76 


77 


87 


82 


78 


77 


78 


88 


83 


80 


79 


80 


3490 


3686 


3681 


3780 


3882 


92 


86 


83 


82 


83 


93 


88 


86 


84 


85 


96 


90 


86 


85 


87 


96 


91 


88 


87 


89 


98 


93 


90 


89 


90 


99 


94 


91 


90 


92 


3601 


96 


93 


92 


94 


03 


98 


96 


94 


96 


04 


99 


96 


96 


97 


3606 


3601 


3698 


3797 


3899 


07 


02 


99 


99 


3901 


09 


04 


3701 


3800 


02 


10 


06 


03 


02 


04 


12 


07 


04 


04 


06 


14 


09 


06 


06 


07 


16 


10 


07 


07 


09 


17 


12 


09 


09 


U 


18 


14 


11 


11 


13 


20 


16 


13 


12 


14 


3621 


3617 


3714 


3814 


3916 


23 


18 


16 


16 


18 


26 


20 


17 


17 


19 


26 


22 


19 


19 


21 


28 


23 


21 


21 


22 


29 


25 


22 


22 


25 


31 


26 


24 


24 


26 


32 


28 


26 


26 


28 


34 


30 


27 


27 


30 


36 


31 


29 


29 


32 


3637 


3633 


3731 


3831 


3933 


39 


34 


32 


32 


35 


40 


36 


34 


34 


37 


42 


38 


36 


36 


38 


43 


39 


37 


38 


40 


45 


41 


39 


39 


42 


47 


43 


41 


41 


44 


48 


44 


42 


43 


46 


60 


46 


44 


44 


47 


61 


47 


46 


46 


49 


3553 


3649 


3747 


3848 


3951 


66 


61 


49 


49 


62 


66 


62 


50 


51 


54 


68 


64 


62 


53 


66 


69 


55 


64 


64 


58 


61 


57 


55 


66 


59 


62 


69 


67 


58 


61 


64 


60 


59 


60 


63 


66 


62 


60 


61 


64 


67 


64 


62 


63 


66 



13968 
70 
71 
73 
75 
77 
78 
80 
82 
84 

3985 
87 
89 
91 
92 
94 
96 
98 
99 

4001 

m 

06 
06 
10 
12 
14 
15 
17 
19 

4021 
22 
24 
26 
28 
29 
31 
33 
35 
37 

4038 
40 
42 
44 
46 
47 
49 
51 
62 
54 

4056 
58 
60 
61 
63 
66 
67 
69 
70 
72 



TABLE VII. 



96 



M- 


660 


67© 


680 


690 


60O 


610 


620 


630 640 


650 


660 


670 


680 


690 





4074 


418314294 


4409^4627 


4649 


4775 


4906 5039 


5179 


5324 


5474 


5631 


6796 


1 


76 


84 


96 


11 


29 


61 


77 


07, 42 


81 


26 


77 


33 


97 


2 


77 


86 


96 


13 


31 


63 


79 


09 44 


84 


28 


79 


36 


6800 


3 


79 


88 


4300 


16 


33 


66 


81 


12* 46 


86 


31 


82 


39 


03 


4 


81 


90 


02 


17 


36 


67 


84 


14< 49 


88 


33 


84 


42 


06 


6 


83 


92 


04 


19 


37 


60 


86 


16, 51 


91 


36 


87 


44 


09 


6 


86 


94 


06 


21 


39 


62 


88 


18 53 


93 


38 


89 


47 


11 


7 


B6 


96 


06 


23 


41 


64 


90 


20 56 


95 


41 


92 


60 


14 


8 


B8 


97 


09 


26 


43 


66 


92 


23 


68 


98 


43 


95 


52 


17 


9 


90 


99 


11 


27 


45 


68 


94 


26 


60 


5200 


46 


97 


66 


20 


10 


4092 


4201 


4313 


4429 


4647 


4670 


4796 


4927 5062 


5203 


5348 


6600 


6668 


6823 


11 


94 


03 


16 


31 


49 


72 


98 


29 


66 


06 


51 


02 


60 


26 


12 


96 


06 


17 


33 


61 


74 


4801 


31 


67 


07 


63 


05 


63 


28 


13 


97 


07 


19 


34 


63 


76 


03 


34 


69 


10 


66 


07 


66 


31 


14 


99 


06 


21 


36 


66 


78 


05 


36 


71 


12 


58 


10 


68 


34 


16 


4101 


10 


23 


38 


67 


80 


07 


38 


74 


14 


61 


13 


71 


37 


16 


03 


12 


26 


40 


69 


82 


09 


40 


76 


17 


63 


16 


74 


39 


17 


04 


14 


27 


42 


62 


84 


11 


43 


78 


19 


66 


18 


76 


42 


18 


06 


16 


28 


44 


64 


87 


14 


46 


81 


22 


68 


20 


79 


46 


19 


08 


18 


30 


46 


66 


89 


16 


47 


83 


24 


71 


23 


82 


48 


20 


4110 


4220 


4332 


4448 


4568 


4691 


4818 


4949 


6086 


5226 


6373 


5526 


5686 


6861 


21 


12 


21 


34 


60 


70 


93 


20 


61 


88 


29 


76 


28 


87 


64 


22 


13 


23 


36 


62 


72 


96 


22 


54 


90 


31 


78 


31 


90 


66 


23 


16 


26 


38 


64 


74 


97 


24 


66 


92 


34 


80 


33 


93 


69 


24 


17 


27 


40 


66 


76 


99 


26 


68 


95 


36 


83 


36 


96 


62 


26 


19 


29 


42 


68 


78 


4701 


29 


60 


97 


38 


86 


39 


98 


66 


26 


21 


31 


44 


60 


80 


03 


31 


63 


99 


41 


88 


41 


5701 


68 


27 


22 


32 


46 


62 


82 


06 


33 


66 


5102 


43 


90 


44 


04 


71 


28 


24 


34 


47 


64 


84 


07 


36 


67 


04 


46 


93 


46 


06 


74 


29 


26 


36 


49 


66 


86 


10 


37 


69 


06 


48 


95 


49 


99 


76 


30 


4128 


4238 


4361 


4468 


4688 


4712 


4839 


4972 


6106 


5260 


5396 


5562 


5712 


6879 


31 


30 


40 


63 


70 


90 


14 


42 


74 


11 


63 


5401 


64 


16 


82 


32 


32 


42 


66 


72 


92 


16 


44 


76 


13 


66 


03 


67 


17 


85 


33 


33 


44 


67 


74 


94 


18 


46 


78 


16 


68 


06 


69 


20 


88 


34 


36 


46 


69 


76 


96 


20 


48 


81 


18 


60 


08 


62 


23 


91 


36 


37 


47 


61 


78 


98 


22 


60 


83 


20 


63 


11 


66 


26 


94 


36 


39 


^ 


63 


80 


4600 


24 


62 


86 


22 


66 


13 


67 


28 


96 


37 


41 


66 


82 


02 


26 


65 


87 


26 


67 


16 


70 


31 


99 


38 


42 


6» 


67 


84 


04 


28 


57 


90 


27 


70 


18 


73 


34 


5902 


39 


44 


66 


69 


86 


06 


31 


69 


92 


29 


72 


21 


76 


36 


06 


40 


4146 


4267 


4370 


4488 


4608 


4733 


4RB1 


4994 


5132 


6276 


5423 


5678 


6739 


6908 


41 


48 


69 


72 


90 


10 


36 


63 


96 


34 


77 


26 


80 


42 


11 


42 


60 


60 


74 


92 


12 


37 


66 


99 


36 


80 


28 


83 


45 


14 


43 


62 


62 


76 


94 


14 


39 


68 


6001 


39 


82 


31 


86 


47 


17 


44 


63 


64 


78 


96 


16 


41 


70 


03 


41 


84 


33 


88 


60 


19 


46 


66 


66 


80 


97 


18 


43 


72 


06 


43 


87 


36 


91 


63 


22 


46 


57 


68 


82 


99 


20 


46 


74 


08 


46 


89 


38 


94 


66 


26 


47 


69 


70 


84 


4501 


23 


47 


76 


10 


48 


92 


41 


96 


68 


28 


48 


61 


72 


86 


03 


26 


60 


79 


12 


61 


94 


43 


99 


61 


31 


49 


62 


74 


88 


06 


27 


62 


81 


14 


53 


97 


46 


5602 


64 


34 


60 


4164 


4276 


4390 


4607 


4629 


4754 


4983 


5017 


5166 


6299 


5448 


5604 


6767 


5937 


61 


66 


77 


92 


09 


31 


66 


86 


19 


68 


5301 


51 


07 


70 


40 


62 


66 


79 


94 


11 


33 


66 


87 


21 


60 


04 


64 


10 


72 


43 


63 


70 


81 


96 


13 


36 


60 


90 


23 


62 


06 


66 


12 


76 


46 


64 


72 


83 


98 


16 


37 


62 


92 


26 


66 


09 


69 


16 


78 


48 


66 


73 


86 


99 


17 


39 


64 


94 


28 


67 


11 


61 


17 


81 


51 


66 


76 


87 


4401 


19 


41 


66 


96 


30 


69 


14 


64 


20 


83 


64 


W 


77 


89 


03 


21 


43 


69 


98 


33 


72 


16 


66 


23 


86 


67 


68 


79 


91 


06 


23 


46 


71 


4901 


36 


74 


19 


69 


26 


89 


60 


69 


81 


92 


07 


26 


47 


73 


03 


37 


76 


21 


71 


28 


92 


f 



■MM 



96 



TABLE VII. 



M. 70O 710 720 



981 
1916201 
22 O&i 



730 


740 


6634 6746*, 


3R 


49 


41 


63 


46 67 


48 60 


62 


64 


66 


68 


68 


71 


62 


76 


66 


79 

cron 



72| 86 

76 90 

79 93 

83 97 
86 6801 

90: 04 

93 08 

96 12 

6600| 16 

6603 6819 
071 23 



10 
14 
17 
21 
24 
28 
31 
36 



26 
30 
34 
38 
41 
45 
49 
63 



66396866 



60 
64 
68 
71 



42 

46 
49 
63 
66; 75 
60, 79 
63 83 
67! 86 
70, 90 



6467 6674 
70 77 
73 81 
86 
88 
92 
95 
99 



6894 
98 
01 
05 
09 
13 
17 
20 



6702] 24 
06| 28 

67106932 

13 36 
17 40 
20 43 



24 
28 
31 
35 
38 
42 



47 
61 
65 
59 
63 
66 



76° 

6970 
74 
78 
82 
86 
90 
94 
97 

7001 
05 

7009 
13 
17 
21 
25 
29 
33 
37 
41 
46 

7049 
62 
56 
60 
64 
68 
72 
76 
80 
84 

7068 
92 
96 

7100 
04 
08 
12 
16 
20 
24 

7128 
32 
36 
40 
45 
49 
53 
57 
61 
65 

7169 
73 
77 
81 
85 
89 
94 
98 

7202 
06 



760 

7210 
14 
18 
22 
27 
31 
35 
39 
43 
47 

7252 
66 
60 
64 

68 
73 
77 
81 
85 
89 

7294 
98 

7382 
06 
11 
15 
19 
23 
28 
32 

7336 
40 
46 
49 
63 
58 
62 
66 
71 
75 

7379 
84 
88 
92 
97 
01 
06 
10 
14 
19 

7423 
27 
32 
36 
41 
45 
49 
54 
58 
63 



770 

7467 
72 
76 
81 
86 
90 
94 
98 

7603 
07 

7612 
16 
21 
26 

30 
35 
39 
44 
48 
53 

7557 
62 
66 
71 
76 
80 
85 
89 
94 
98 

7603 
08 
12 
17 
22 
26 
31 
36 
40 
45 

7650 
54 
59 
64 
68 
73 
78 
83 
87 
92 

7697 
7702 
06 
11 
16 
21 
25 
30 
35 
40 



78<» 

7745 
49 
64 
69 
64 
69 
74 
78 
83 



790 

8046 
51 
56 
61 
67 
72 
77 
83 
88 



88 93 

77938099 
988104 



7803 
06 
13 
17 
22 
27 
32 



09 
15 
20 
25 
31 
36 
41 



37! 47 



7842 
47 
62 
67 
62 
67 
72 
77 
82 
87 

7892 
97 

7902 
07 
12 
17 
22 
27 
32 
37 

7942 
48 
63 
58 
63 
68 
73 
78 
83 
89 

7994 
99 

8004 
09 
14 
20 
25 
30 
35 
40 



8O0 

8375 
81 
87 
93 
98 

8404 
10 
16 
22 
27 

8433 
39 
45 
61 

671 

63 

69 

74 

80 

86 



8152 8492 
58! 98 
638604 
68 101 



74 
79 
85 
90 
96 
8201 



16 
22 
28 
34 
40 
46 



8207,8552 
12 58 
18 64 
23, 71 
29: n 
34 83 
40, 89 
45! 95 
518601 
66, 07 

8262 8614 

67| 20 

73! 26 

79" 32 

84l 38 

90, 44 

95 51 

83011 57 

07| 63 

12, 69 

8318 8676 
24 82 
29 88 
36 93 
418701 
47! 07 
52 14 
58 20 
64 26 
69. 33 



8I0 

8739 
46 
62 
58 
66 
71 
78 
84 
91 
97 

8804 
10 
17 
23 
30 
36 
43 
49 
66 
63 



820 

19145 
53 
60 
67 
74 
82 
89 
96 

9203 
11 

9218 
26 
33 
40 
48 
56 
62 
70 
77 
85 



8869 9292 
7&9300 



83 
89 
96 
8903 
09 
16 
23 
30 

8936 
43 
50 
57 
63 
70 
77 
84 
91 
98 



07 
15 
22 
30 
38 
46 
53 
60 

9368 
76 
83 
91 
99 
07 
14 
22 
30 
38 



9005 9445 
12 53 



18 
25 
32 
39 
46 



61 
69 
77 
85 
93 



63 9501 
60| 09 

67 17 



9074 9525 

81 1 33 

88 41 

96' 49 

9103; 57 

10, 66 

17 73 

24 81 

31| 89 

38- 98 



830 

9606 
14 
22 
31 
39 
47 
66 
64 
72 
81 

9689' 
97 

9706 
14 
23 
31 
40 
48 
57 
65 

9774 
83 
91 

9800 
09 
17 
26 
36 
44 
52 

9861 
70 
79 
88 
97 

9906 
15 
24 
33 
42 

9951 

60 

69 

78 

87 

9996 

10005 

10015 

10024 

10033 

10043 
10052 
10061 
10071 
10080 
10089 
10099 
10108 
10118 
10127 



A TABLE 



OF 



ATMOSPHERICAL REFRACTIONS, 



WITH CORRECTIONS 



FOB THE HSIOHT OP 



THE BAROMETER AND THERMOMETER, 



TAKEN FROM THE NAUTICAL ALMANAC. 



98 











TABLE Vm. 










TABLE OF REFRACTIONS. 








JM-- 


J>^' 


m 


Dig. 


1 JMA". 


DH- 


Dig. 


Dig. 


jSSuU, 


A-.30. 
Tk. 50(». 




f 1 B. 




M^id€» 


ITr. SO. 
7%.50o. 


VJM, 


.ft. 




\ 6 


f 1 62 


2:i 


24.1' 


1.7« 


10 6 


ih 


.6 


/ // 

10.8 


.& 


10 


11 30 


2.1 


23.4 


1.64 


10 


5 15 


.6 


10.6 


.67 


20 


11 10 


2.0 


22.7 


1.68 


20 


5 10 


.5 


10.4 


.66 


30 


10 50 


1.9 


22.0 


1.63 


30 


5 5 


.6 


10.2 


.64 


40 


10 32 


1.8 


21.3 


1.48 


40 


6 


.5 


10.1 


.63 


50 


10 15 


1.7 


20.7 


1.43 


50 


4 56 


.4 


9.9 


.62 


5 


956 


1.6 


2a6 


1.38 


11 


4 61 


.4 


9.8 


.60 


10 


9 42 


1.6 


19.1 


1.84 


10 


4 47 


.4 


9.6 


.59 


20 


927 


1.6 


19.1 


1.30 


20 


4 43 


A 


9.5 


.68 


30 


9 11 


1.4 


18.6 


1.26 


30 


439 


.4 


9.4 


.67 


40 


8 58 


1.3 


18.1 


1.22 


40 


4 35 


.4 


9.2 


.56 


60 


846 


1.3 


17.6 


1.19 


60 


4 31 


.4 


9.1 


.55 


6 


8 32 


1.2 


17.2 


1.15 


12 


4 2ai 


.38 


9.00 


.556 


10 


8 20 


1.2 


16.8 


1.11 


10 


4 24.4 


.37 


8.86 


.548 


20 


8 9 


1.1 


16.4 


1.09 


20 


4 20.8 


.36 


8.74 


.541 


30 


7 58 


1.1 


16.0 


1.06 


30 


4 17.3 


.35 


8.63 


.533 


40 


7 47 


1.0 


16.7 


1.03 


40 


4 13.9 


.33 


8.51 


.524 


60 


7 37 


1.0 


15.3 


1.00 


50 


4 10.7 


.32 


8.41 


.517 


7 


7 27 


l.O 


15.0 


.98 


13 


4 7.5 


.31 


8.30 


.609 


10 


7 17 


.9 


14.6 


.95 


10 


4 4.4 


.31 


8.20 


.503 


20 


7 8 


.9 


14.3 


.93 


20 


4 1.4 


.30 


8.10 


.496 


30 


6 59 


.8 


14.1 


.91 


30 


3 58.4 


.30 


8.00 


.490 


40 


6 61 


.8 


13.8 


.89 


40 


3 56.5 


.29 


7.89 


.482 


50 


6 43 


.8 


13.6 


.87 


60 


3 52.6 


.29 


7.79 


.476 


8 


6 35 


.7 


13.3 


.86 


14 


3 49.9 


.28 


7.76 


.469 


10 


6 28 


.7 


13.1 


.83 


10 


3 47.1 


.28 


7.61 


.464 


20 


6 21 


.7 


12.8 


.82 


20 


3 44.4 


.27 


7.52 


.458 


30 


6 14 


.7 


12.6 


.80 


30 


3 41.8 


.26 


7.43 


.453 


40 


6 7 


.7 


12.3 


.79 


40 


3 39.2 


.26 


7.34 


.448 


50 


6 


.6 


12.1 


.77 


50 


3 36.7 


.25 


7.26 


.444 


9 


654 


.6 


11.9 


.76 


15 


3 34.3 


.24 


7.18 


.439 


10 


6 47 


.6 


11.7 


.74 


30 


3 27.3 


.22 


6.95 


.424 


20 


5 41 


.6 


11.5 


.73 


16 


3 20.6 


.21 


6.73 


.411 


30 


536 


.6 


11.3 


.71 


30 


3 14.4 


.20 


6.61 


.399 


40 


5 30 


5 


11.1 


.71 


17 


3 8.6 


.19 


6.31 


.386 


50 


5 25 


.5 


ll.O 


.70 


30 


3 2.9 


.18 


6.12 


.374 












18 


2 57.6 


.17 


6.98 


.362 












19 


2 47.7 


.16 


5.61 


.340 



99 



TABLE VIII. 



TABLE OF REFRACTIOKS. 






•rf82 
•079 
•076 
•073 
•070 

•067 
•065 
•062 
•060 
•057 

•055 
•062 
•050 
•047 
•046 

•043 
•040 
•039 
•036 
•033 

•031 
•029 
•027 
•025 
•023 

•021 
•018 
•016 
•014 
•012 

•OlO" 
•008 
•006 
•004 
•002 



App. 
JUitude. 



o 

20 

21 

22 

23 

24 

25 
26 
27 
28 
29 

30 
31 
32 
33 
34 

35 
36 
37 
38 
39 

40 
41 
42 
43 
44 

45 
46 
47 
48 
49 

50 
51 
52 
53 
54 



Refr. 
Br. aO. 
Th.50o. 



/ 
2 
2 
2 
2 
2 

2 
1 
1 
1 
1 

1 
1 
1 
1 
1 

1 
1 
1 
1 
1 

1 
1 
1 
1 
1 



// 
38-7 
30-5 
23-2 
16-5 
101 

4-2 

58-8 
53-8 
49-1 
44-7 

40-5 
36-6 
330 
29-5 
261 

230 
200 
17-1 
14-4 
11-8 

9-3 
6-9 
4-6 
2-4 
0-3 

58-1 
661 
54-2 
52-3 
50-5 

48^8 
47-1 
45-4 
43-8 
42-2 



IHff. 

for 

V Alt. 



// 
•15 
•13 
•12 
•11 
•10 

•09 
•09 
•08 
•08 
•07 

•07 
•06 
•06 
•06 
•05 

•05 
•05 
•05 
•05 
•04 

•04 

•04 

•038 

•03G 

•03^ 

:034 
•033 
•032 
•031 
•030 

•029 
•028 
•027 
•026 
•026 



Dig. 

for 

+ 1 Bar, 



// 
5^3l 
504 
4-79 
4-57 
4-35 

4-16 
3-97 
3-81 
3-65 
3-50 

3-36 
3-23 
3^11 
2-99 
2-88 

2-78 
2-68 
2-58 
2-49 
2-40 

232 
2-24 
2-16 
209 
202 

1-94 
1-88 
181 
175 
r69 

1-63 
1-58 
152 
147 
141 



Dijf 



for __ 

-^l^Fakl Altitude. 



// 

•322 
•305 
•290 
•276 
•264 

•252 
•241 
•230 
•219 
•209 

•201 
•193 
•186 
•179 
•173 

•167 
•161 
•155 
•149 
•144 

•139 
•134 
•130 
•125 
•120 

•117 
•112 
•108 
•104 
•101 

•097 
•094 
•090 
•088 
•085 



App. 



o 

55 

56 

57 

58 

69 

60 
61 
62 
63 
64 

65 
66 
67 
68 
69 

70 
71 
72 
73 
74 

75 
76 

77 
78 
79 

80 
81 
82 
83 
84 

85 
86 
87 
88 
89 



Refr, 
Pr. 30. 
TA.50O. 



40-8 
39-3 
37-8 
36-4 
350 

33-6 
323 
310 
29-7 
28^4 

272 
25-9 
24^7 
23-5 
22-4 

21-2 
19-9 
18^8 
17-7 
16*6 

15-5 
14^4 
134 
12-3 
112 

10-2 
9-2 
8^2 
71 
61 

5-1 
41 
31 
2^0 
1-0 



Dijf, 

for 

V AlU 



•d-25 
•025 
•025 
•024 
•024 

•023 
022 
■022 
•021 
•021 

•020 
•020 
•020 
•020 
•020 

•020 
•020 
•019 
•018 
•018 

■018 
•018 
•017 
•017 
•017 

•017 
•017 
■0J7 
•017 
•017 

•017 
•017 
•017 
•017 
•017 



Dilf. 

for 

HBar. 



V24 
\'3l 
126 
1-22 
117 

1.12 

1.08 

1.04 

.99 

.95 

.91 
.87 
.83 
.79 
.75 

•71 
•67 
•63 
•59 
•56 

•52 
•48 
•45 
•41 
•38 

•34 
•31 

•27 
•24 
•20 

•17 
•14 
•10 
•07 
•03 



% 


of the 


Halflh. 


D,p. 








0'59" 










4 


1 B6 








2 33 


e 


2 44 


9 








11 


3 10 


12 


3 19 


13 




14 


336 


IS 


3 42 


16 


360 














20 


4 IT 






23 


430 


23 


436 


24 


4 42 


















40 


6 4 


45 


627 


GO 


646 


fiO 


7 25 


70 


8 1 






90 


9 6 


lOO 


935 



8ua'Bpsr.inAlt.| 










10 




20 


8 




8 










66 




60 












7S 




80 








90 






Ibe vertical Bemi- 
dUm. of ©or©, 
on account of Re- 
[yaciion. 


Alt. 


Dim. of 

seniHlia. 


6 
7 

e 


19 
11 
11 


10 

12 
13 


8 
7 

6 


16 
16 


4 
4 

3 


18 
M 
30 

46 


3 

2 

1 



"r.'rs.i 


All. 


0" 




10 


3 


15 




20 








30 


S 




9 






46 




BO 


2 










70 


G 


80 






6 



RediidlionofilioCSEqiiflloriEi 








figure of the Earih. 




.J 


Horizon (al Parallai. 




B4' 1 66' 1 58' 1 60- | 62' 




S 


nil 


flO 


nil 


1! 


nli 




fl 


11 '^ 


ir^ 


(IV 


0« 










OFl 




ill 






?0 


i;i 


in 


1,4 


14 


in 




H 


IN 


m 


11 


■/o 


'n 








'J'S 


'?fi 


?fi 




















Sfi 


'U 


HM 


■1(1 


41 


i'i 










4H 


fill 


rtt 










5fi 






















'fi 


67 


71) 


7? 


74 


7(i 












fi^ 




















H4 


H7 


m 


M4 


97 


inn 






11 


IK 


KHi 


li'i 


10 A 












119 






7H 




1(1.6 


Kin 




117 






10.7 


II 1 


MR 


19 


I'^n 




90 


lfl,8 111.2 


11.6 


Z-O 


12.4 





.^^ 



Of^ 



-■^.-y 






fe 



.1 



s 



v. I 




y. .•■ 



^